Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves Shinichi Mochizuki and Shota Tsujimura June 16, 2023 Abstract In this paper, we prove that arbitrary hyperbolic curves over p-adic lo- cal fields admit resolution of nonsingularities [“RNS”]. This result may be regarded as a generalization of results concerning resolution of nonsingu- larities obtained by A. Tamagawa and E. Lepage. Moreover, by combining our RNS result with techniques from combinatorial anabelian geometry, we prove that an absolute version of the geometrically pro-Σ Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields, where Σ denotes a set of prime numbers of cardinality 2 that contains p, holds. This settles one of the major open questions in anabelian geometry. Furthermore, we prove again by applying RNS and combinatorial an- abelian geometry that the various p-adic versions of the Grothendieck- Teichmüller group that appear in the literature in fact coincide. As a corollary, we conclude that the metrized Grothendieck-Teichmüller group is commensurably terminal in the Grothendieck-Teichmüller group. This settles a longstanding open question in combinatorial anabelian geometry. Contents Introduction 2 Notations and Conventions 11 1 Local construction of Artin-Schreier extensions in the special fiber 15 2020 Mathematics Subject Classification: Primary 14H30; Secondary 14H25. Keywords and phrases: anabelian geometry; resolution of nonsingularities; abso- lute Grothendieck Conjecture; combinatorial anabelian geometry; Grothendieck-Teichmüller group; étale fundamental group; tempered fundamental group; hyperbolic curve; configuration space. 1 2 Resolution of nonsingularities for arbitrary hyperbolic curves over p-adic local fields 26 3 Point-theoreticity, metric-admissibility, and arithmetic cuspi- dalization 72 References 103 Introduction Let p be a prime number; Σ a nonempty subset of the set Primes of prime numbers. For a connected noetherian scheme S, we shall write Π S for the étale fundamental group of S, relative to a suitable choice of basepoint. For any field F of characteristic 0, any field extension F E, and any algebraic variety [i.e., a separated, geometrically integral scheme of finite type] Z over F , we shall def write Z E = Z × F E and denote by F an algebraic closure [well-defined up to def isomorphism] of F and by G F = Gal(F /F ) the absolute Galois group of F . For any field F of characteristic 0 and any algebraic variety Z over F , we shall write (Σ) def Π Z = Π Z /Ker(Π Z F  Π Σ Z F ), where Π Z F  Π Σ Z F denotes the maximal pro-Σ quotient. Here, we recall that (Σ) Π Z is often referred to as the geometrically pro-Σ fundamental group of Z. We shall write Q p for the field of p-adic numbers; C p for the p-adic completion of Q p . We shall refer to a finite extension field of Q p as a p-adic local field. For any hyperbolic curve Z over either the algebraic closure of a mixed character- istic complete discrete valuation field of residue characteristic p or the p-adic completion of such an algebraic closure, we shall write Π tp Z for the Σ-tempered fundamental group of Z, relative to a suitable choice of base- point [cf. the subsection in Notations and Conventions entitled “Fundamental groups”]. If S 1 , S 2 are schemes, then we shall write Isom(S 1 , S 2 ) for the set of isomorphisms of schemes between S 1 and S 2 . If G 1 , G 2 are profinite groups, then we shall write OutIsom(G 1 , G 2 ) for the set of isomorphisms of profinite groups, considered up to composition with an inner automorphism arising from an element G 2 . In the present paper, we give a complete affirmative answer to the following question: 2 Does an arbitrary hyperbolic curve over a p-adic local field admit resolution of nonsingularities? Before continuing, we review the notion of resolution of nonsingularities. Let K (⊆ K) be a mixed characteristic complete discrete valuation field of residue characteristic p; v a valuation on a field F that contains K. Write O K for the ring of integers of K; O v for the ring of integers determined by v; m v O v for the maximal ideal of O v . Then we shall say that v is a p-valuation [over K] [cf. Definition 2.2, (i)] if O K = O v K [which implies that p m v ]. Here, the phrase “over K” will be omitted in situations where the base field K is fixed throughout the discussion. We shall say that v is residue-transcendental def [cf. Definition 2.2, (i)] if it is a p-valuation whose residue field k v = O v /m v is a transcendental extension of the residue field of K. Let Z be a hyperbolic curve over K. Then we shall say that Z satisfies Σ-RNS [i.e., “Σ-resolution of nonsingularities” cf. Definition 2.2, (vii); [Lpg1], Definition 2.1] if the following condition holds: Let v be a discrete residue-transcendental p-valuation on the func- tion field K(Z) of Z. Then there exists a connected geometrically pro-Σ finite étale Galois covering Y Z such that Y has stable reduction [over its base field], and v coincides with the restriction [to K(Z)] of a discrete valuation on the function field K(Y ) of Y that arises from an irreducible component of the special fiber of the stable model [cf. Definition 2.1, (vi)] of Y . If Z is an O K -scheme, then we shall write Z s for the special fiber of Z [i.e., the fiber of Z over the closed point of Spec O K ]. Let Z be an O K -scheme. Then [cf. Definition 2.1, (i), (ii), (iv)]: (i) We shall say that Z is a compactified model of Z over O K if Z is a proper, flat, normal scheme over O K whose generic fiber is the [uniquely deter- mined, up to unique isomorphism] smooth compactification of Z over K. (ii) Suppose that the cusps of Z are K-rational. Then we shall say that Z is a compactified stable model of Z over O K if Z is a compactified model of Z over O K such that the following conditions hold: the geometric special fiber of Z is a semistable curve [i.e., a reduced, connected curve each of whose nonsmooth points is an ordinary dou- ble point]; the images of the sections Spec O K Z determined by the cusps of Z [which we shall refer to as cusps of Z] lie in the smooth locus of Z and do not intersect each other. Z, together with the cusps of Z, determines a pointed stable curve. Then one verifies immediately, by considering blow-ups of compactified models at specified closed points [cf. also Remark 2.2.2; Proposition 2.4, (iii), (iv)], that, if Σ is a set of cardinality 2 that contains p [cf. the situation discussed 3 in Theorem A below], then the condition of satisfying Σ-RNS discussed above is in fact equivalent to the following property, which clarifies the meaning of “resolution of nonsingularities”: Let Z be a compactified model of Z over O K ; z Z s a closed point. Then, after possibly replacing K by a suitable finite extension field of K, there exist a connected geometrically pro-Σ finite étale Galois covering Y Z of hyperbolic curves over K, a compactified stable model Y of Y over O K , a morphism Y Z of compactified models over O K that re- stricts to the finite étale Galois covering Y Z, an irreducible component D of Y s whose normalization is of genus 1, and whose image in Z s is z Z s . This alternative formulation of the condition of satisfying Σ-RNS is useful to keep in mind when considering the relationship between Theorem A below and the following result due to A. Tamagawa [cf. [Tama2]], which played an important role in motivating the following result due to E. Lepage [cf. [Lpg1]]: Suppose that Σ = Primes, that the residue field of K is algebraic over the finite field of cardinality p, and that Z is the compactified stable model of Z over O K . Then there exist a connected finite étale Galois covering Y Z and an irreducible component D of Y s as in the above alternative formulation [cf. [Tama2], Theorem 0.2, (v)]. Suppose that Σ = Primes, that K is a p-adic local field, and that Z is a hyperbolic Mumford curve over K. Then Z satisfies Σ-RNS [cf. [Lpg1], Theorem 2.7]. Our first main result may be regarded as a generalization of these results [cf. Theorem 2.17]: Theorem A (Resolution of nonsingularities for arbitrary hyperbolic curves over p-adic local fields). Suppose that Σ Primes is a subset of cardinality 2 that contains p, and that K is a p-adic local field. Let X be a hyperbolic curve over K; L a mixed characteristic complete discrete valuation field of residue characteristic p that contains K as a topological subfield. Then X L satisfies Σ-RNS if and only if the residue field of L is algebraic over the finite field of cardinality p. In the remainder of the present Introduction, we discuss various anabelian applications/consequences of Theorem A. First, by applying a certain sophisticated version of the argument applied in the proof of [Tsjm], Theorem 2.2, we obtain the following consequence of Theorem A concerning the determination of closed points on arbitrary hyperbolic curves via geometric tempered fundamental groups, which generalizes [Tsjm], Theorem 2.2 [cf. Corollary 2.5, (ii), which in fact applies to hyperbolic curves over more general p-adic fields; Remark 2.5.1]: 4 Theorem B (Determination of closed points on arbitrary p-adic hy- perbolic curves by geometric tempered fundamental groups). Suppose that Σ Primes is a subset of cardinality 2 that contains p. Let X , X  X , X  X for the univer- be hyperbolic curves over Q p . Write X , Π tp , respectively. Let sal geometrically pro-Σ coverings corresponding to Π tp X X x X (C p ), x X (C p ). Write X x (respectively, X x ) for the hyperbolic curve X C p \{x } (respectively, X C p \{x }) over C p . Let σ  : Π tp Π tp x x X X be an isomorphism of topological groups that fits into a commutative diagram Π tp −−−−→ Π tp X X σ  x x   Π tp −−−−→ Π tp , X X σ where the vertical arrows are the natural surjections [determined up to com- position with an inner automorphism] induced by the natural open immersions X x → X C p , X x → X C p of hyperbolic curves; the lower horizontal arrow σ is the isomorphism of topological groups [determined up to composition with an inner automorphism] induced by a(n) [uniquely determined] isomorphism σ X : X X of schemes over Q p . Then x = σ X (x ). Next, we consider applications of Theorem A to Grothendieck Conjecture- type results in anabelian geometry. We begin by recalling the following question, which may may be considered as an absolute version of the Grothendieck Con- jecture for hyperbolic curves over p-adic local fields: Let X , X be hyperbolic curves over p-adic local fields. Then is the natural map Isom(X , X ) −→ OutIsom(Π X , Π X ) bijective? This question may be regarded as one of the major open questions in anabelian geometry. In this context, we recall that, in the case of the relative version of the Grothendieck Conjecture for arbitrary hyperbolic curves, many satisfactory results have been obtained [cf. [PrfGC], Theorem A; [Tama1], Theorem 0.4; [LocAn], Theorem A]. In particular, the first author of the present paper gave a complete affirmative answer to the original question posed by A. Grothendieck [i.e., the original “Grothendieck Conjecture”] in quite substantial generality [cf. [LocAn], Theorem A; [AnabTop], Theorem 4.12]. On the other hand, in the case 5 of the absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields [i.e., the question of the above display], analogous results had not been obtained previously, due to the existence of outer isomor- phisms of the absolute Galois groups of p-adic local fields that do not arise from isomorphisms of fields [cf., e.g., [NSW], the Closing Remark preceding Theorem 12.2.7]. In this direction, in some sense the strongest known result, prior to the present paper, was the following result [cf. [AbsTopII], Corollary 2.9]: Suppose that Σ Primes is a subset of cardinality 2 that contains p. Let X , X be hyperbolic curves over p-adic local fields. Write (Σ) (Σ) (Σ) (Σ) OutIsom D X , Π X ) OutIsom(Π X , Π X ) for the subset determined by the isomorphisms that induce bijections between the respective sets of decomposition subgroups associated to the closed points of X and X . Then the natural map Isom(X , X ) −→ OutIsom D X , Π X ) (Σ) (Σ) is bijective. Moreover, in [AbsTopII], [AbsTopIII], the first author developed a technique, called “Belyi cuspidalization”, that allows one to reconstruct the decomposition subgroups associated to the closed points of strictly Belyi-type hyperbolic curves and proved that an absolute version of the Grothendieck Conjecture holds for such curves [cf. [AbsTopIII], Theorem 1.9]. In the present paper, we apply Theorem A, together with some combinatorial anabelian geometry, to reconstruct the set of C p -valued points of a hyperbolic curve over Q p from its geometric tempered fundamental group [cf. Corollary 3.10, which in fact applies to hyperbolic curves over more general p-adic fields]: Theorem C (Reconstruction of C p -valued points via geometric tem- pered fundamental groups). Suppose that Σ Primes is a subset of cardi-  X nality 2 that contains p. Let X be a hyperbolic curve over Q p . Write X tp  may be for the universal pro-Σ covering corresponding to Π X [so Gal( X/X) tp  identified with the pro-Σ completion of Π X ]. Then the set X(C p ) equipped  hence also, by passing to the set of with its natural action by Gal( X/X)   p )  X(C p ) may be reconstructed, Gal( X/X)-orbits, the quotient set X(C in a purely combinatorial/group-theoretic way and functorially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp X . Note that it follows immediately from Theorem C that, under the assump- tion that Σ Primes is a subset of cardinality 2 that contains p, one may reconstruct the set of closed points, hence also the set of associated de- composition subgroups, of a hyperbolic curve over a p-adic local field, in a purely combinatorial/group-theoretic way and functorially with respect to iso- morphisms of topological groups, from its geometrically pro-Σ étale fundamental group [cf. the proof of Theorem 3.11 for more details]. Here, it is also interesting to observe that 6 this reconstruction of the set of decomposition subgroups asso- ciated to closed points from the geometrically pro-Σ étale funda- mental group of a hyperbolic curve over a p-adic local field may be thought of as a sort of a weak version of the Section Conjecture. In particular, by applying [AbsTopII], Corollary 2.9, together with some com- binatorial anabelian geometry, we obtain a complete affirmative answer to the question considered above, i.e., an absolute version of the Grothendieck Con- jecture for arbitrary hyperbolic curves, as well as for the configuration spaces associated to such hyperbolic curves, over p-adic local fields [cf. Theorems 3.12; 3.13]: Theorem D (Absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields). Suppose that Σ Primes is a subset of cardinality 2 that contains p. Let X , X be hyperbolic curves over p-adic local fields. Then the natural map Isom(X , X ) −→ OutIsom(Π X , Π X ) (Σ) (Σ) is bijective. Theorem E (Absolute version of the Grothendieck Conjecture for con- figuration spaces associated to arbitrary hyperbolic curves over p-adic local fields). Let X , X be hyperbolic curves over p-adic local fields; n , n positive integers. Write X n (respectively, X n ) for the n -th (respectively, n - th) configuration space associated to X (respectively, X ). Then the natural map Isom(X n , X n ) −→ OutIsom(Π X , Π X ) n n is bijective. In the context of the relationship between the theory developed in the present paper and the Section Conjecture, it is of interest to note the following conse- quences of the theory underlying Theorem C [cf. Proposition 2.4, (vii); Propo- sition 3.5, (iii); Proposition 3.9, (iv) all of which in fact apply to hyperbolic curves over more general p-adic fields]: Theorem F (Consequences related to the Section Conjecture over p-adic fields). Suppose that Σ Primes is a subset that contains p, and that the residue field of K is an algebraic extension of the finite field of cardinality p. Let l Σ \ {p}; H G K a closed subgroup such that the intersection H I K of H with the inertia subgroup I K of G K admits a surjection to [the profinite group] Z l . Write Ω for the p-adic completion of K; Ω H Ω for the subfield of Ω fixed by H. Then the following properties hold: (i) Let X be a proper hyperbolic curve over K that is in fact defined over a p-adic local field contained [in a fashion compatible with the respective  X a universal geometrically pro-Σ covering, s X : topologies] in K, X 7 (Σ) def  H Π X = Gal( X/X) a section of the restriction to H of the natural (Σ)  an for the topological pro-Berkovich space surjection Π X  G K . Write X associated to [i.e., the inverse limit of the underlying topological spaces  Then of the Berkovich spaces associated to the finite subcoverings of ] X. an  there exists at most one point X that is fixed by the restriction, via (Σ) s X , to H of the natural action of Π X on the topological pro-Berkovich  an ; if, moreover, the restriction to H of the l-adic cyclotomic space X character of K has open image, then there exists a unique such point  an . Finally, s X arises from an Ω H -rational point X(Ω H ) if and X (Σ)  an . only if its image is a maximal stabilizer Π X × G K H of a point X (ii) Suppose that the restriction to H of the l-adic cyclotomic character of K has open image. Let Y , Z be [not necessarily proper!] hyperbolic curves over K that are in fact defined over a p-adic local field contained [in a fashion compatible with the respective topologies] in K; Y  Y , Z  Z universal geometrically pro-Σ coverings of Y , Z, respectively; (Σ) def f : Y Z a dominant morphism over K; s Y : H Π Y = Gal( Y  /Y ), (Σ) def  = Gal( X/X) sections of the restrictions to H of the s Z : H Π Z (Σ) (Σ) respective natural surjections Π Y  G K , Π Z  G K such that s Y is (Σ) mapped, up to Π Z -conjugation, by f to s Z via the map induced by f on geometrically pro-Σ fundamental groups. Then s Y arises from an Ω H - rational point Y H ) if and only if s Z arises from an Ω H -rational point Z(Ω H ). Here, we observe in passing that it follows immediately from Theorem F, (ii) [i.e., as stated above], together with [BSC], Theorem 5.33, that a similar result to the result stated in Theorem F, (ii), holds if the field K is replaced by a number field, but [since global issues over number fields lie beyond the scope of the present paper] we shall not discuss this in detail in the present paper. Next, we recall that one of the key ingredients in the theory of p-adic arith- metic cuspidalizations developed in [Tsjm], §2, is Lepage’s resolution of nonsin- gularities [i.e., [Lpg1], Theorem 2.7], which may be regarded as a special case of Theorem A. Our next result is obtained by applying this theory of p-adic arith- metic cuspidalizations [i.e., [Tsjm], §2], together with some elementary observa- tions concerning the lengths of nodes of stable models of hyperbolic curves [cf. Proposition 3.15], to prove that the various p-adic versions of the Grothendieck- Teichmüller group that appear in the literature [cf. [Tsjm], Remark 2.1.2] in fact coincide and are commensurably terminal in the Grothendieck-Teichmüller group [cf. Theorem 3.16; Corollary 3.17]: Theorem G (Equality and commensurable terminality of various p-adic versions of the Grothendieck-Teichmüller group). Suppose that Σ = def Primes. Write X = P 1 Q \ {0, 1, ∞}; p GT Out(Π X ) 8 for the Grothendieck-Teichmüller group [cf. [CmbCsp], Remark 1.11.1]; GT M GT (⊆ Out(Π X )) for the metrized Grothendieck-Teichmüller group [cf. [CbTpIII], Remark 3.19.2]; def = GT Out(Π tp GT tp p X ) Out(Π X ) [cf. the subsection in Notations and Conventions entitled “Fundamental groups”; [Tsjm], Definition 2.1]. Then the natural inclusion GT M GT tp p of subgroups of GT is an equality. In particular, it holds that GT M = GT p = GT G = GT tp p [cf. [Tsjm], Remark 2.1.2]. Moreover, GT M = GT p = GT G = GT tp p is com- M mensurably terminal in GT, i.e., the commensurator C GT (GT ) of GT M in GT is equal to GT M . Note that the commensurable terminality in the final portion of Theorem G may be regarded as an affirmative answer to the question posed in the discussion immediately preceding Theorem E in [CbTpIII], Introduction. Finally, we conclude with an interesting complement to the theory of p-adic arithmetic cuspidalizations by applying Theorem C, together with the theory of metric-admissibility developed in [CbTpIII], §3, to construct certain p-adic arithmetic cuspidalizations of the geometric tempered fundamental group of a hyperbolic curve over Q p equipped with certain relatively mild auxiliary data [cf. Definition 3.19; Theorem 3.20; Remarks 3.20.1, 3.20.2]. The contents of the present paper may be summarized as follows: In §1, we discuss in detail certain local computations motivated by [Lpg1], Proposition 2.4, but formulated entirely in the language of schemes and for- mal schemes, i.e., without resorting to the use of notions from the theory of Berkovich spaces concerning iterates of the p-th power morphism of the mul- tiplicative group scheme G m over the ring of integers of a mixed characteristic discrete valuation field of residue characteristic p. By restricting such mor- phisms to suitable formal neighborhoods, we conclude that smooth curves of genus 1 appear in the special fibers of suitable models of the domain curves of such morphisms, i.e., as certain Artin-Schreier coverings of curves of genus 0 [cf. Proposition 1.6; Remark 1.6.2]. In §2, we begin by discussing various generalities concerning models of a hyperbolic curve over a mixed characteristic complete discrete valuation field [cf. Definition 2.1; Proposition 2.3]. In particular, we discuss the definition of the notion of Σ-RNS [cf. Definition 2.2, (vii)], together with closely related basic properties of this notion [cf. Propositions 2.4; Corollary 2.5]. Another important 9 notion in this context is the purely combinatorial notion of VE-chains associated to a hyperbolic curve over a mixed characteristic complete discrete valuation field [cf. Definition 2.2, (iii)]. This notion is closely related to the topological Berkovich space associated to the hyperbolic curve [cf. Proposition 2.3, (vii), (viii)]. Finally, we apply certain constructions involving p-divisible groups to extend the Artin-Schreier coverings constructed locally in §1 to coverings of an arbitrary hyperbolic curve over a p-adic local field [cf. Propositions 2.12, 2.13; Theorem 2.16]. This leads naturally to a proof of Theorem A [cf. Theorem 2.17], hence also, by combining Theorem A with the theory of VE-chains developed in the earlier portion of §2, together with some combinatorial anabelian geometry, of Theorem B. In §3, we begin by recalling the well-known classification of the points of the topological Berkovich space associated to a proper hyperbolic curve over a mixed characteristic complete discrete valuation field via the notion of type i points, where i {1, 2, 3, 4}. Next, we introduce a certain combinatorial clas- sification of the VE-chains considered in §2 and observe that this classification of VE-chains leads naturally to a purely combinatorial characterization of the well-known classification via type i points mentioned above. We then apply the theory of §2 to give a group-theoretic characterization, motivated by [but by no means identical to] the characterization of [Lpg2], §4, of the type i points in terms of the geometric Σ-tempered fundamental group of the hyperbolic curve. The theory surrounding this group-theoretic characterization leads naturally to proofs of Theorems C and F. Moreover, by combining this group-theoretic characterization with [AbsTopII], Corollary 2.9; [HMM], Theorem A, we obtain proofs of Theorems D and E [cf. Theorems 3.12; 3.13]. We then switch gears to discuss metric-admissibility for p-adic hyperbolic curves. This discussion of metric-admissibility yields, in particular, a proof of Theorem G and leads naturally to the discussion of p-adic arithmetic cuspidalizations associated to geometric tempered fundamental groups equipped with certain relatively mild auxiliary data [cf. Definition 3.19; Theorem 3.20; Remarks 3.20.1, 3.20.2] men- tioned above. Acknowledgements This research was supported by the Research Institute for Mathematical Sci- ences, an International Joint Usage/Research Center located in Kyoto Univer- sity, as well as the International Center for Research in Next Generation Geom- etry. 10 Notations and Conventions Numbers: The notation Primes will be used to denote the set of prime numbers. The notation N will be used to denote the set of nonnegative integers. The notation Q ≥0 will be used to denote the additive monoid of nonnegative rational numbers. The notation Z will be used to denote the additive group or ring of integers. The notation Q will be used to denote the field of rational numbers. The notation R will be used to denote the field of real numbers. For each x R, the notation x will be used to denote the greatest integer x. If p is a prime number, then the notation Q p will be used to denote the field of p-adic numbers; the notation Z p will be used to denote the additive group or ring of p-adic integers; the notation Q p will be used to denote an algebraic closure of Q p ; the notation C p will be used to denote the p-adic completion of Q p . We shall refer to a finite extension field of Q p as a p-adic local field. Monoids: Let M be a commutative monoid. In this subsection, we regard the set of positive integers N ≥1 as a directed set via its multiplicative structure, i.e., def i j i | j. For i N ≥1 , write M i = M . For i, j N ≥1 such that i j, write φ i,j : M i M j for the homomorphism of monoids obtained by multiplication by ji . Then we shall refer to the inductive limit of the inductive system {M i , φ i,j } on the directed set N ≥1 as the perfection of M . Rings and fields: Let R be a ring. Then we shall write R × for the multiplicative group of units of the ring. Let F be a perfect field, p a prime number. Then the notation F will be def used to denote an algebraic closure of F ; G F = Gal(F /F ). Suppose that F is a valuation field, i.e., a field equipped with a valuation map [cf., e.g., the axioms of [EP], Eq. (2.1.2)]. Then we shall write O F for the ring of integers of F ; m F for the maximal ideal of O F . Thus, the valuation map on F induces an isomorphism of ordered abelian groups between F × /O F × and the value group of the valuation field F . In particular, any valuation map on the field F is determined, up to unique isomorphism [in the evident sense], by the ring of integers F associated to the valuation map. In the present paper, we shall use the term valuation to refer to an isomorphism class of valuation maps on a field F , i.e., the collection of valuation maps that give rise to the same ring of integers F . We shall refer to a specific valuation map within a given isomorphism class of valuation maps on a field as a normalized valuation, i.e., an isomorphism class of valuation maps on the field, together with a specific valuation map [belonging to this class], which we shall refer to as a normalization. If the value group of the 11 valuation field F is isomorphic, as an ordered abelian group, to a subgroup of the underlying additive group of R, then we shall say that F is a real valuation field. If F is a mixed characteristic real valuation field of residue characteristic p, then the notation v p will be used to denote the normalized valuation on F whose normalization is determined by the condition that v p (p) = 1 R. If F is a henselian valuation field of characteristic 0, then we shall write I F G F for the inertia subgroup; F F ur (⊆ F ) for the maximal unramified extension. If F is a real henselian valuation field of characteristic 0, then we shall write F  ur for the completion of F ur [cf. Remark 2.2.4 for more details]. Topological groups: Let G be a topological group; H G a closed subgroup of G. Then we shall write G ab for the abelianization of G; C G (H) for the commensurator of H G, i.e., def C G (H) = {g G | H g · H · g −1 is of finite index in H and g · H · g −1 }. We shall say that the closed subgroup H is commensurably terminal in G if H = C G (H). Let Σ Primes be a nonempty subset. Then we shall write G Σ for the pro-Σ completion of G. We shall write Aut(G) for the group of continuous automorphisms of the topological group G, Inn(G) Aut(G) for the subgroup of inner automorphisms def of G, and Out(G) = Aut(G)/Inn(G). Suppose that G is center-free. Then we have a natural exact sequence of groups 1 −→ G ( Inn(G)) −→ Aut(G) −→ Out(G) −→ 1. If J is a group, and ρ : J Out(G) is a homomorphism, then we shall denote by out G  J the group obtained by pulling back the above exact sequence of groups via ρ. Thus, we have a natural exact sequence of groups out 1 −→ G −→ G  J −→ J −→ 1. Suppose further that the topology of G admits a countable basis consisting of characteristic open subgroups of G. Then one verifies immediately that the topology of G induces a natural topology on the group Aut(G), hence on the group Out(G). In particular, one verifies easily that if J is a topological group, out and ρ : J Out(G) is continuous, then G  J admits a natural topological group structure. Let G 1 , G 2 be profinite groups. Then we shall write OutIsom(G 1 , G 2 ) 12 for the set of isomorphisms of profinite groups, considered up to composition with an inner automorphism arising from an element G 2 . Semi-graphs: Let Γ be a connected semi-graph [cf. [SemiAn], §1]. Then we shall refer to the dimension over Q of the first homology module of Γ [with coefficients in Q] H 1 (Γ, Q) as the loop-rank of Γ. We shall say that Γ is untangled if every closed edge of Γ abuts to two distinct vertices. Schemes: Let K be a field; K L a field extension; X an algebraic variety [i.e., a separated, geometrically integral scheme of finite type] over K. Then we shall def write X L = X × K L. Suppose that X is a smooth proper curve [i.e., a smooth, proper algebraic variety of dimension 1] over K. Then we shall write J(X) for the Jacobian of X. Let p be a prime number; A a semi-abelian variety or a p-divisible group over K. Then we shall write T p A for the p-adic Tate module associated to [the p-power torsion points valued in some fixed algebraic closure of K of] A. Suppose that K is a valuation field. Let X be an O K -scheme. Then we shall write X s for the special fiber of X [i.e., the fiber of X over the closed point of Spec O K ]. Let S 1 , S 2 be schemes. Then we shall write Isom(S 1 , S 2 ) for the set of isomorphisms of schemes between S 1 and S 2 . Curves: We shall use the term “hyperbolic curve” [i.e., a family of hyperbolic curves over the spectrum of a field] as defined in [MT], §0. We shall use the term “n-th configuration space” as defined in [MT], Definition 2.1, (i). Log schemes: If X log is a fine log scheme, then we shall write X for the underlying scheme of X log ; M X for the étale sheaf of monoids on X that defines the log structure of X log ; × , which we shall refer to as the characteristic of X log [cf. M X = M X /O X [MT], Definition 5.1, (i)]. def Let K be a complete discrete valuation field; Y a hyperbolic curve over K; Y a def compactified semistable model of Y over O K [cf. Definition 2.1, (ii)]. Write S = Spec O K ; S log for the log scheme determined by the log structure associated to the closed point of S. Then it follows immediately from [Hur], §3.7, §3.8, 13 that the multiplicative monoid of sections of O Y that are invertible on [the open subscheme of Y determined by] Y determines a natural log structure on Y. Denote the resulting log scheme by Y log . Then one verifies immediately that the natural morphism of schemes Y S extends to a a proper, log smooth morphism Y log S log of fine log schemes. Let y be a geometric point of Y s . Write M y pf for the perfection of the stalk of the characteristic M Y at y. Then one verifies immediately that, if the image of y in Y s is a smooth point that is not a cusp (respectively, a cusp; a node), then M y pf Q ≥0 (respectively, M y pf Q ≥0 × Q ≥0 ; M y pf Q ≥0 × Q ≥0 ). Fundamental groups: For a connected noetherian scheme S, we shall write Π S for the étale funda- mental group of S, relative to a suitable choice of basepoint. Let Σ Primes be a nonempty subset; K a perfect field; X an algebraic variety over K. Then we shall write def Δ X = Π X K ; (Σ) def Π X = Π X /Ker(Δ X  Δ Σ X ), Σ where Δ X  Δ Σ X denotes the natural surjection. We shall refer to Δ X (respec- (Σ) tively, Π X ) as the geometric pro-Σ fundamental group (respectively, geometri- cally pro-Σ fundamental group) of X. Let p be a prime number, Σ Primes a nonempty subset. Suppose that K is a mixed characteristic complete discrete valuation field of residue characteristic p. Write Ω for the p-adic completion of K. Let F be one of the [topological] fields K, K, and Ω; X a hyperbolic curve over F . If F = Ω, then we shall write Π tp X for the Σ-tempered fundamental group of X, i.e., in the terminology of [CbTpIII], Definition 3.1, (ii), the “Σ-tempered fundamental group” of the smooth log curve over F determined by X [where we note that one verifies immediately that the field  K of loc. cit. may be taken to to be the field F = Ω of the present discussion, and that the discussion of loc. cit. may be applied to the situation of the present discussion even if the X of the present discussion does not descend to the field  K” of loc. cit.]. Here, we recall that, when F = Ω, it follows immediately from [André], Proposition 4.3.1 [and the surrounding discussion], that the universal topological coverings of finite coverings corresponding to char- acteristic open subgroups of Π tp X of finite index determine a countable collection tp of characteristic open subgroups of Π tp X that form a basis of the topology of Π X , tp hence determine a natural topology on Out(Π X ). Thus, if F = K, then we obtain a natural continuous outer action G K Out(Π tp X Ω ) −→ 14 and hence, since Π tp X Ω is center-free [cf., e.g., [CbTpIII], Proposition 3.3, (i), (ii); [MT], Proposition 1.4], a topological group Π tp X def = out Π tp X Ω  G K , which we refer to as the geometrically Σ-tempered fundamental group of X. Moreover, if F = K, then Π tp X is equipped with a natural continuous surjection tp Π X  G K , whose kernel [which in fact may be naturally identified with Π tp X Ω ] we def = Π tp denote by Δ Σ-tp X X Ω and refer to as the geometric Σ-tempered fundamental group of X. If F = K, then, after possibly replacing K by a finite extension of K, we may assume that X descends to a hyperbolic curve X K over K [so that X may be naturally identified with “(X K ) K ”], and we shall refer to Π tp X def = Π tp X Ω as the Σ-tempered fundamental group of X. Finally, we note that, in the case where F = K, it follows immediately from [CanLift], Proposition 2.3, (ii) [cf. also [CbTpIII], Proposition 3.3, (i)], together with the definition of the Σ- tp tempered fundamental group, that the pro-Σ completion of Π tp X = Π X Ω may be naturally identified, for suitable choices of basepoints, with the geometric pro-Σ fundamental group Δ Σ X of X. 1 Local construction of Artin-Schreier extensions in the special fiber Let p be a prime number. In the present section, we perform various local computations concerning iterates of the p-th power morphism of the multiplica- tive group scheme G m over the ring of integers of a mixed characteristic discrete valuation field of residue characteristic p. As a consequence, by restricting such morphisms to suitable formal neighborhoods, we conclude that smooth curves of genus 1 appear in the special fibers of suitable models of the domain curves of such morphisms [cf. Proposition 1.6; Remark 1.6.2]. This observation will be applied in §2 to prove that arbitrary hyperbolic curves over p-adic local fields ad- mit resolution of nonsingularities. The contents of this section may be regarded as an alternative and somewhat more detailed discussion of [Lpg1], Proposition 2.4, that is phrased entirely in the language of schemes and formal schemes and does not resort to the use of Berkovich spaces. First, we begin with several elementary lemmas concerning the p-adic valu- ations of the coefficients of certain polynomials and power series [cf. Lemmas 1.1, 1.2, 1.3]. 15 Lemma 1.1. Let K be a mixed characteristic discrete valuation field of residue characteristic p. Suppose that K contains a primitive p-th root of unity ζ p K. def Write π = 1 ζ p K;   def f (x) = π −p (1 + πx) p 1 K[x], where x denotes an indeterminate. Then it holds that f (x) (x p x) m K [x] O K [x]. def Proof. First, write c = p + (−π) p−1 . Then since π = 1 ζ p K, it holds that c = p + (−π) p−1 = p + (−π) p−1 + π −1 ((1 π) p 1). Observe that these equalities imply that v p (c) > 1. In particular, it holds that v p (p + (−π) p−1 ) > 1, hence that v p (p + π p−1 ) > 1. Next, observe that, if we write  f (x) = a i x i K[x], 1≤i≤p then since v p (p) = 1 = v p p−1 ), and v p (p + π p−1 ) > 1, it holds that a 1 = p , π p−1 a p = 1, v p (a 1 + 1) > 0, v p (a i ) > 0 (∀i = 1, p). Thus, we conclude that f (x) (x p x) m K [x]. This completes the proof of Lemma 1.1. Lemma 1.2. Let K be a mixed characteristic discrete valuation field of residue characteristic p; n a positive integer. Write  q i x i K[[x]] f (x) = 1 + i≥1 where x denotes an indeterminate for the n-th root of 1+x K[[x]] whose constant term is equal to 1. Then it holds that v p (q 1 ) = −v p (n), v p (q i ) −iv p (n) v p (i!) −i v p (n) + 1 p 1 (∀i 2) i [cf. the well-known elementary fact that v p (i!) p−1 ]. If, moreover, n is prime to p, then v p (q i ) 0 for each positive integer i. 1 Proof. First, we observe, by considering the Taylor expansion of (1 + x) n at 0, that 1 1 k q i = i! n 0≤k≤i−1 16 for each positive integer i. The equality and inequalities of the second display of the statement of Lemma 1.2 follow immediately. Next, suppose that n is prime to p. For each positive integer m, write def Q m K for the O K -subalgebra generated by {q j } 1≤j≤m−1 . Write Q 0 = O K . Then, by comparing the coefficients of x m in the left- and right-hand sides of the equality n  1+ q i x i = 1 + x, i≥1 we conclude that nq m Q m−1 . In particular, since n is prime to p, it holds that q m Q m−1 . Thus, by induction, we conclude that q i O K for each positive integer i. This completes the proof of Lemma 1.2. Lemma 1.3. Let K be a mixed characteristic discrete valuation field of residue characteristic p; n a positive integer;  a i s i O K [[s]] \ {1}, g(s) = 1 + i≥1 where s denotes an indeterminate. Write i 0 for the smallest positive integer i such that a i = 0; 0 s for the O K -valued point of Spec O K [[s]] obtained by mapping s → 0. Then the following hold: (i) Suppose that v p (a i ) 2v p (n) for each i 1. Then g(s) admits an n-th root  1+ b i s i O K [[s]]. i≥1 1 (ii) Suppose that there exists an element x O K such that v p (x) = 3i 0 (p−1) . Then there exist a positive integer j, 2 an element b O K satisfying the inequalities v p (b) 1 and 3(p−1) 1 v p (b) v p (b) < p−1 , and an isomorphism h : O K [[s]] O K [[s]] of topological O K -algebras such that h maps 0 s → 0 s and h(g(x j s)) = 1 + bs i 0 . (iii) Suppose that g(s) = 1+a i 0 s i 0 , where a i 0 satisfies the inequalities v p (a i 0 ) def 1 2 1 and 3(p−1) v p (a i 0 ) v p (a i 0 ) < p−1 . Write μ = v p (a i 0 ) 1 0. μ Then g(s) admits a p -th root  1 c i 0 i s i 0 i O K [[s]], g(s) = 1 + i≥1   p 2 2 v p (c i 0 ) < p−1 , v p (c 2i 0 ) 2 1 + 3(p−1) 1, and where 1 + 3(p−1)   1 v p (c i 0 i ) i 1 3(p−1) for each positive integer i > 2. 17 (iv) In the notation of (iii), suppose that K contains a primitive p-th root of unity ζ p K and an element c K such that c i 0 = π p , c i 0 def p where we write π = 1 ζ p . [Note that since v p (c i 0 ) < p−1 , it holds that v p (c) > 0.] Write  1 d i 0 i s i 0 i = π −p (g(cs) pm 1) K[[s]]. def i≥1 p , v p (d 2i 0 ) > Then it holds that d i 0 = 1, v p (d i 0 ) = 0 > v p (c i 0 ) p−1 p 2i 0 sup{v p (c ) p−1 , 0}, and v p (d i 0 i ) > i · sup 1 1 p , v p (c i 0 ) 0 3(p 1) p 1 for each positive integer i > 2. Moreover, v p (d i 0 i ) > v p (c i 0 i ) for each sufficiently large positive integer i. Proof. First, we verify assertion (i). Note that g(s) admits an n-th root  1+ b i s i K[[s]]. i≥1 Thus, it suffices to verify that v p (b i ) 0 for each positive integer i. Note that 1 if v p (n) 1 (≥ p−1 ), then 2iv p (n) i v p (n) + 1 p 1 0.  Thus, by applying Lemma 1.2 [where we take “x” to be the element i≥1 a i s i ], together with our assumption that v p (a i ) 2v p (n) for each positive integer i, we conclude that v p (b i ) 0 for each positive integer i. This completes the proof of assertion (i). Next, we verify assertion (ii). Fix an element x O K such that v p (x) = def ij 1 3i 0 (p−1) . For each pair of positive integers (i, j), write b i,j = x a i . Note that it follows immediately from our assumption on v p (x) [i.e., by thinking of the real 1 line R modulo integral multiples of 3(p−1) = i 0 v p (x)] that there exists a positive integer j such that v p (b i 0 ,j ) 1, 2 1 3(p−1) v p (b i 0 ,j ) v p (b i 0 ,j ) < p−1 , and 18 v p (b i,j ) v p (b i 0 ,j ) + 2v p (i 0 ) for each positive integer i > i 0 . def Fix such a positive integer j and write b = b i 0 ,j . Then the existence of an isomorphism h : O K [[s]] O K [[s]] of topological O K -algebras such that h maps 0 s → 0 s and h(g(x j s)) = 1 + bs i 0 follows immediately from Lemma 1.3, (i), where we take “n” to be i 0 and “g(s)” to be g (s) = b −1 s −i 0 (g(x j s) 1) def 1 def [so g(x j s) = 1+bs i 0 g (s)]. [That is to say, h is defined by taking h(s(g (s)) n ) = 1 s, where “(g (s)) n denotes the n-th root of Lemma 1.3, (i).] This completes the proof of assertion (ii). Assertion (iii) follows immediately from Lemma 1.2 [applied to x = a i 0 s i 0 ], together with the elementary fact that v p (i!) 1 for i = 2. Finally, we verify assertion (iv). It follows immediately from the various definitions involved that p(i−1) d i 0 i = c i 0 i c −i . i 0 π 1 In particular, it holds that d i 0 = 1. Moreover, since v p (π) = p−1 , and v p (c i 0 ) < p [cf. Lemma 1.3, (iii)], it holds that p−1 v p (d i 0 i ) = v p (c i 0 i ) iv p (c i 0 ) + p(i 1) p > v p (c i 0 i ) . p 1 p 1 Suppose that i = 2 (respectively, i 3). Then it follows immediately from Lemma 1.3, (iii), that v p (c i 0 i ) 2 p 2 1+ p 1 3(p 1) 1 p 1 = > 0 p 1 3(p 1) (respectively, v p (c i 0 i )− 1 1 2p 4 p p p i 1− 3 1− = 0). p 1 3(p 1) p 1 3(p 1) p 1 p 1 Thus, we conclude that v p (d 2i 0 ) > 0, and v p (d i 0 i ) > i 1 1 3(p 1) p 0 p 1 for each positive integer i > 2. The remainder of assertion (iv) follows immedi- ately from the inequalities already obtained, together with the inequalities p 1 1 p > inf 1 , + p 1 3(p 1) 2 · 3(p 1) 2(p 1) p 2 1 = 1+ > 3(p 1) p 1 3(p 1) v p (c i 0 ) [cf. Lemma 1.3, (iii)]. This completes the proof of assertion (iv), hence of Lemma 1.3. 19 Definition 1.4. Let K be a mixed characteristic discrete valuation field of residue characteristic p; c m K \ {0}. Then we shall write φ c : B c −→ Spec O K [[t]] where t denotes an indeterminate for the blow-up of Spec O K [[t]] with center given by the closed subscheme defined by the ideal (c, t); b c B c for the generic point of the exceptional irreducible component of the special fiber of B c [i.e., the fiber of B c over the closed point of Spec O K ]; U c = Spec O K [[t]][s c ]/(cs c t) B c where s c denotes an indeterminate, but, by a slight abuse of notation, we shall also use the notation “s c to denote the element of Γ(U c , O U c ) determined by the indeterminate “s c ”— for the open subscheme obtained by removing the strict transform of the special fiber of Spec O K [[t]]. Note that it follows immediately from the various definitions involved that φ c induces a morphism  c def = Spec O K [[s c ]] −→ Spec O K [[t]] φ  c : U over O K that maps t to cs c . Proposition 1.5 (Local construction of Artin-Schreier extensions in the special fiber I). We maintain the notation of Definition 1.4. Suppose that K contains a primitive p-th root of unity ζ p K. Write k for the residue def field of K; π = 1 ζ p K; f : Spec O K [[t]] −→ Spec O K [[t]] for the [manifestly] finite flat morphism over O K corresponding to the homo- morphism of topological O K -algebras that maps t → (1+t) p −1. Then f induces a finite morphism f ˜ : B π −→ B π p over O K that maps b π → b π p and induces a finite flat morphism U π U π p , whose induced morphism on special fibers is the morphism induced on spectra by the injective homomorphism k[s π p ] −→ k[s π ] over k that maps s π p → s s π . Proof. First, observe that we have inclusions of ideals in O K [[t]] 2 (π, t) p +p p , (1 + t) p 1) (π, t). Indeed, the inclusion p , (1 + t) p 1) (π, t) is immediate. Next, we verify the 2 def inclusion (π, t) p +p p , (1 + t) p 1). Write = (1 + t) p 1. Then it holds 20 2 2 2 that p t p + (p 2 ) t p + p ). In particular, it holds that t p ( p , π p ) 2 p , t̃). Thus, we conclude that π i t p +p−i p , t̃) for each integer i such that 2 0 i p 2 + p, hence that (π, t) p +p p , (1 + t) p 1). This completes the proof of the observation. The observation of the preceding paragraph, together with Lemma 1.1, im- plies, by the definition of the blow-up, that the morphism f functorially induces a [proper and quasi-finite, hence] finite morphism f ˜ : B π B π p over O K , which, in turn, induces a [manifestly] finite flat morphism U π −→ U π p over O K such that t → (1 + t) p 1,   s π p → π −p (1 + πs π ) p 1 [cf. Lemma 1.1]. Indeed, it follows from Lemma 1.1 that this morphism U π U π p determines a dominant morphism Spec k[s π ] −→ Spec k[s π p ] [between open subschemes of the respective exceptional irreducible components of the special fibers of B π , B π p ] that corresponds to the injective homomorphism k[s π p ] −→ k[s π ] over k that maps s π p → s s π . This completes the proof of Proposition 1.5. Remark 1.5.1. In the notation of Proposition 1.5, write G m = Spec O K [u, u 1 ] for the multiplicative group scheme over O K , where u denotes an indeterminate; ι : Spec O K [[t]] −→ G m for the morphism that corresponds to the homomorphism over O K that maps u → 1 + t. Then we have a commutative diagram U π f ˜ |  −−−−→ Spec O K [[t]] −−−−→ G m ι φ π | U π p  f  U π p −−−−−→ Spec O K [[t]] −−−−→ G m , φ π p | U π p ι where the right-hand vertical arrow denotes the p-th power morphism; the verti- cal arrows are finite flat morphisms of degree p [cf. Proposition 1.5]; the second square is cartesian; the first square is cartesian, up to taking the normalization of the fiber product that would make the first square “truly cartesian”. 21 Proposition 1.6 (Local construction of Artin-Schreier extensions in the special fiber II). We maintain the notation of Remark 1.5.1. Let  a i t i O K [[t]] \ {1}. g(t) = 1 + i≥1 Write i 0 for the smallest positive integer i such that a i = 0; λ g : Spec O K [[t]] −→ Spec O K [[t]] for the morphism over O K corresponding to the homomorphism of topological O K -algebras that maps t → g(t) 1. Then the following hold: (i) After possibly replacing K by a suitable finite field extension of K, there exist a positive integer μ, elements c 1 , c 2 m K \ {0}, an isomorphism λ h : Spec O K [[s c 1 ]] Spec O K [[s c 1 ]] over O K , and a morphism ξ g : Spec O K [[s c 1 ]] −→ G m over O K satisfying the following conditions: Write 0 c 1 for the O K -valued point of Spec O K [[s c 1 ]] obtained by map- ping s c 1 → 0. Then λ h maps 0 c 1 → 0 c 1 , and ξ g maps 0 c 1 to the identity element of G m (O K ). There exists a commutative diagram Spec O K [[s c 1 ]] −−−−→ Spec O K [[t]] −−−−→ Spec O K [[t]]  c φ 1  λ h  λ g ι  Spec O K [[s c 1 ]] −−−−→ ξ g G m −−− μ −→ p G m , where the right-hand lower horizontal arrow denotes the p μ -th power morphism. Write τ : Spec O K [[t]] Spec O K [[s c 1 ]] for the isomorphism over O K corresponding to the isomorphism of topological O K -algebras that maps s c 1 → t; η(s c 2 ) O K [[s c 2 ]] for the image of u via the homomorphism O K [u, u 1 ] O K [[s c 2 ]] in- duced by the composite Spec O K [[s c 2 ]] −→ Spec O K [[t]] −→ Spec O K [[s c 1 ]] −→ G m .  c φ 2 τ 22 ξ g Then it holds that η(s c 2 ) 1 m K [[c 2 s c 2 ]], and, moreover, π −p (η(s c 2 ) 1) s i c 0 2 m K [s c 2 ] + m K [[c 2 s c 2 ]] = m K [s c 2 ] + m K [[t]]. In particular, there exists a morphism θ g : U c 2 −→ U π p over O K that fits into the following commutative diagram U c 2 −−−−−→ Spec O K [[t]] −−−−→ Spec O K [[s c 1 ]] τ φ c 2 | U c 2 θ g  ξ g  U π p −−−−−→ Spec O K [[t]] −−−−→ φ π p | U π p ι G m . (ii) Fix a collection of data (μ, c 1 , c 2 , λ h , ξ g ) as in (i). Write def Y = U c 2 × U π p U π for the fiber product determined by the morphism θ g : U c 2 U π p and the morphism U π U π p induced by f ˜ [cf. Proposition 1.5]. Then the natural morphism Y s −→ (U c 2 ) s = Spec k[s c 2 ] induced by the first projection morphism Y U c 2 corresponds to the nat- ural injective homomorphism k[s c 2 ] → k[s c 2 , y]/(y p y s i c 0 2 ) over k, where y denotes an indeterminate. Proof. First, we consider assertion (i). We begin by applying Lemma 1.3, (ii), where we take “g(s)” to be g(t) [i.e., so the indeterminate “s” corresponds to t], and we observe that, by replacing K by a suitable finite extension of K, we may assume without loss of generality that there exists an “x” as in Lemma 1.3, (ii). This yields an isomorphism “h : O K [[s]] O K [[s]]” as in Lemma 1.3, (ii), whose induced morphism on spectra where we interpret the indeterminate “s” to be s c 1 , and we take “x j to be c 1 [so “x j s” corresponds to c 1 s c 1 = t] we take to be λ h . Here, we recall that this isomorphism “h” of Lemma 1.3, (ii), satisfies a condition “h(g(x j s)) = 1 + bs i 0 ”. Next, we would like to apply Lemma 1.3, (iii), where we take “1 + a i 0 s i 0 to be the “1 + bs i 0 of Lemma 1.3, (ii) [i.e., so the indeterminate “s” still corresponds to s c 1 ]. This yields a 1 power series “g(s) as in Lemma 1.3, (iii). We then take the “μ” of Lemma 1.3, (iii), to be μ and define ξ g to be the morphism over O K corresponding 1 to the homomorphism that maps u to this power series “g(s) [i.e., where the indeterminate “s” still corresponds to s c 1 ]. This yields a collection of data 23 (μ, c 1 , λ h , ξ g ) that satisfies the first two itemized conditions of Proposition 1.6, (i). The third [and final] itemized condition of Proposition 1.6, (i), now follows by translating the various estimates of Lemma 1.3, (iv), into the notation of the present situation, where we take the “c” of Lemma 1.3, (iv), to be c 2 , and we observe that, again by replacing K by a suitable finite extension of K, we may assume without loss of generality that there exist “ζ p and “c” as in Lemma 1 1.3, (iv). Also, we observe that the power series “g(cs) of Lemma 1.3, (iv), corresponds to η(s c 2 ) [i.e., where the indeterminate “s” corresponds to s c 2 ]. This completes the proof of assertion (i). Next, we verify assertion (ii). Recall that the morphism θ g : U c 2 U π p corresponds to the homomorphism of topo- logical O K -algebras O K [[t]][s π p ]/(π p s π p t) −→ O K [[t]][s c 2 ]/(c 2 s c 2 t) that maps s π p → π −p (η(s c 2 ) 1), t → η(s c 2 ) 1, while the morphism U π U π p corresponds to the homomorphism of topological O K -algebras O K [[t]][s π p ]/(π p s π p t) −→ O K [[t]][s π ]/(πs π t) that maps s π p → π −p ((1 + πs π ) p 1), Then since and t → (1 + t) p 1. π −p ((1 + πs π ) p 1) (s s π ) m K [[s π ]] π −p (η(s c 2 ) 1) s i c 0 2 m K [s c 2 ] + m K [[t]] [cf. Lemma 1.1; Proposition 1.6, (i)], it holds that the morphism (U c 2 ) s (U π p ) s corresponds to the homomorphism k[s π p ] −→ k[s c 2 ] over k that maps s π p → s i c 0 2 , while the morphism (U π ) s (U π p ) s corresponds to the homomorphism k[s π p ] −→ k[s π ] over k that maps s π p → s s π . 24 Thus, we conclude that the first projection morphism Y s = (U c 2 ) s × (U πp ) s (U π ) s (U c 2 ) s = Spec k[s c 2 ] corresponds to the natural injective homomorphism k[s c 2 ] → k[s c 2 , y]/(y p y s i c 0 2 ) over k. This completes the proof of assertion (ii), hence of Proposition 1.6. Remark 1.6.1. In the notation of Proposition 1.6, we observe that the first projection morphism Y U c 2 fits into a commutative diagram Y −−−−→ U π θ Y f Y  f ˜ |  −−−−→ Spec O K [[t]] −−−−→ G m ι φ π | U π p  f  U c 2 −−−−→ U π p −−−−−→ Spec O K [[t]] −−−−→ G m , θ g φ π p | U π p ι where the first vertical arrow f Y denotes the first projection morphism Y U c 2 ; the left-hand upper horizontal arrow θ Y denotes the second projection morphism Y U π ; the vertical arrows are finite flat morphisms of degree p [cf. Proposition 1.5, Remark 1.5.1]; the first and third squares are cartesian; the second square is cartesian, up to taking the normalization of the fiber product that would make the second square “truly cartesian”. Remark 1.6.2. Let k be a field of characteristic p; n a positive integer. Write C for the Artin-Schreier curve over k defined by the equation y p y = x n , where x and y are indeterminates; g C for the genus of C. Then it follows immediately from Hurwitz’s formula that g C = (n  1)(p 1) , 2 where n  denotes the greatest positive integer that divides n and is prime to p. In particular, if n is not a power of p, then g C 1. [Indeed, by considering the Frobenius morphism, one reduces immediately to the case where n = n  . Moreover, the computation of g C is immediate when n = 1 in light of the form of the equation y p y = x. Thus, the computation of g C reduces to the computation of the genus of a tamely ramified cyclic covering of the projective line of degree n whose ramification consists solely of p+1 totally ramified points.] 25 2 Resolution of nonsingularities for arbitrary hy- perbolic curves over p-adic local fields Let p be a prime number. In the present section, we apply certain con- structions involving p-divisible groups to extend the Artin-Schreier coverings constructed locally in §1 to coverings of an arbitrary hyperbolic curve. As a consequence, we prove that arbitrary hyperbolic curves over p-adic local fields satisfy RNS, i.e., “resolution of nonsingularities” [cf. Definition 2.2, (vii); The- orem 2.17]. This result may be regarded as a generalization of results obtained by A. Tamagawa and E. Lepage [cf. [Tama2], Theorem 0.2; [Lpg1], Theorem 2.7]. Historically [cf., e.g., the discussion in the Introduction to [Tama2]], the roots of these results of Tamagawa and Lepage may be traced back to the tech- nique of “passing to a covering with singular reduction of a given curve with smooth reduction over a p-adic local field” applied in the proof of [PrfGC], The- orem 9.2 [cf. also Proposition 2.3, (xii), below]. Moreover, the techniques of [AbsTopII], §2, may be regarded as a sort of weak, pro-p version of Tamagawa’s RNS [cf. [AbsTopII], Remark 2.6.1]. In fact, the approach of the present sec- tion may be regarded as a sort of amalgamation of the techniques of [Lpg1] with the techniques of [AbsTopII], §2. At any rate, from a historical point of view, it is interesting to observe how various RNS results have been motivated by and indeed are deeply intertwined with various results in anabelian geometry [cf. Corollary 2.5, as well as Theorems 3.12, 3.13 in §3 below]. First, we begin by fixing our conventions concerning models of hyperbolic curves [cf. [DM], Definition 1.1; [Knud], Definition 1.1]. Definition 2.1. Let K be a valuation field; X a hyperbolic curve over K; X a scheme over O K . Then: (i) We shall say that X is a compactified model of X over O K if X is a proper, flat, normal scheme of finite presentation over O K whose generic fiber is the [uniquely determined, up to unique isomorphism] smooth compactifi- cation of X over K. (ii) Suppose that the cusps of X are K-rational. Then we shall say that X is a compactified semistable model of X over O K if X is a compactified model of X over O K such that the following conditions hold: the geometric special fiber of X is a semistable curve [i.e., a reduced, connected curve each of whose nonsmooth points is an ordinary dou- ble point]; the images of the sections Spec O K X determined by the cusps of X [which we shall refer to as cusps of X ] lie in the smooth locus of X and do not intersect each other. Suppose that X is a compactified semistable model of X over O K . Then we shall say that X has split reduction if X s is split [i.e., each of the irreducible components and nodes of X s is geometrically irreducible]. 26 (iii) Suppose that X is a compactified semistable model of X over O K that has split reduction. Let L be a finite extension of K equipped with a valuation that extends the valuation on K; X a compactified semistable model of X L over O L such that X has split reduction and dominates X . Then we shall say that X is a toral compactified semistable model relative to X if each irreducible component of X s that maps to a closed point of X s [via the uniquely determined morphism X X ] is normal of genus 0 and has precisely 2 nodes. Suppose that X is a toral compactified semistable model relative to X . Let v be a vertex of the dual graph associated to X s . Then we shall say that v is a toral semistable vertex of X if v corresponds to an irreducible component of X s that maps to a closed point of X . (iv) We shall say that X is a compactified stable model of X over O K if X is a compactified semistable model of X over O K such that X , together with the cusps of X , determines a pointed stable curve. (v) We shall say that X is a semistable model of X over O K if X is obtained by removing the cusps from a [uniquely determined, by Zariski’s Main Theorem, up to unique isomorphism] compactified semistable model of X over O K . Suppose that X is a semistable model of X over O K . Then we shall say that X has split reduction if X s is split [i.e., each of the irreducible components and nodes of X s is geometrically irreducible]. (vi) We shall say that X is a stable model of X over O K if X is obtained by removing the cusps from a [uniquely determined, by Zariski’s Main Theorem, up to unique isomorphism] compactified stable model of X over O K . (vii) We shall say that X has stable reduction over K if there exists a [necessarily unique, up to unique isomorphism] stable model of X over O K . Suppose that X is a stable model of X over O K . Then we shall say that X has split stable reduction over K if X s is split [i.e., each of the irreducible components and nodes of X s is geometrically irreducible]. Remark 2.1.1. It follows from elementary commutative algebra/scheme theory [cf. [EGAIV 2 ], Corollaire 6.1.2; [EGAIV 3 ], Proposition 12.1.1.5] that any com- pactified model as in Definition 2.1, (i), is of dimension 2 whenever K is a complete discrete valuation field. Remark 2.1.2. In the notation of Definition 2.1, suppose that X is a proper hyperbolic curve over K. Then it follows immediately from the various defini- tions involved that the notion of a compactified semistable model of X over O K coincides with the notion of a semistable model of X over O K . 27 Remark 2.1.3. In the notation of Definition 2.1, suppose that K is a complete discrete valuation field, and that X has split stable reduction over K. Let X be a compactified semistable model of X over O K that has split reduction; φ : Y X a morphism of compactified semistable models over O K that re- stricts to a connected finite étale covering Y X over K. Then one verifies immediately by considering the map induced on irreducible components of the respective special fibers by the necessarily finite, hence surjective morphism [induced by Y X ] between the [two-dimensional, normal, integral] spectra of the completions of the local rings of X , Y at the closed points under consider- ation that φ always maps a smooth closed point of Y that is isolated in the fiber of φ to a smooth closed point of X . Remark 2.1.4. In the notation of Remark 2.1.3, let X be a toral compactified semistable model relative to X ; e an edge of the dual graph associated to X s ; b a branch of the node e; v a toral semistable vertex of X that maps to [i.e., for which the corresponding irreducible component maps to the node corresponding  e of the local ring of X at e is isomorphic to to] e. Recall that the completion O O K [[x, y]]/(xy a), where a m K \ {0}, and x, y denote indeterminates chosen so that the ideal (x) (⊆ O K [[x, y]]/(xy a)) corresponds to b. Observe that this ideal (x) is independent of the choice of x, y [cf. [Hur], §3.7, Lemma]. In particular, we obtain a homomorphism of local rings  v ψ : O K [[x, y]]/(xy a) −→ O  v denotes the completion of the local ring of X at the generic point where O of the irreducible component of X corresponding to v. Write ord v (−) for the  v whose normalization is determined by normalized valuation associated to O the condition that ord v (p) = 1 R. Thus, we obtain a rational number def 0 < ρ b,v = ord v (ψ(x)) < 1 ord v (ψ(a)) associated to b and v, which is in fact independent of the normalization of “ord v (−)”. If, moreover, we write b  for the other branch of e, then one verifies immediately that ρ b,v + ρ b  ,v = 1. Finally, we observe that given any rational number ρ such that 0 < ρ < 1, there exist, after possibly replacing K by a suitable finite extension field of K, an X and v as above such that ρ b,v = ρ. Indeed, it follows immediately from the theory of pointed stable curves, as ex- posed in [Knud], that, by possibly replacing K by a suitable finite extension field of K and X by a suitable dense open subscheme of X, we may assume that X is the [unique, up to unique isomorphism] compactified stable model of X over O K , and that ρ may be written as a fraction whose denominator divides the positive integer v p (a). Then it follows again from the theory of pointed stable curves, as exposed in [Knud], that, if we take 28 X to be the [unique, up to unique isomorphism] compactified stable model over O K of the hyperbolic curve obtained by removing from X a suitable K-rational point of X with center at e [cf. the construction of the displayed r  X ,x →) homomorphism “( O O K [[s, t]]/(st π K ) O K in the portion of the proof of Proposition 2.3, (iii), below, concerning the case where “x is a nonsmooth closed point of X ”] and v to be the unique irreducible component of X s that maps to a closed point of X s via the natural morphism X X [cf. the morphism “X [x] X in the portion of the proof of Proposition 2.3, (iii), below, concerning the case where “x is a nonsmooth closed point of X ”], then ρ b,v = ρ, as desired. Remark 2.1.5. In the notation of Remark 2.1.4, we observe that, for a fixed choice of b, the assignment v → ρ b,v Q that assigns to a toral semistable vertex v of X that maps to e the rational number ρ b,v is injective. Indeed, it follows immediately from the theory of pointed stable curves, as ex- posed in [Knud] i.e., by adding finitely many suitably positioned cusps and then considering the various contraction morphisms that arise from eliminating cusps that, to verify the asserted injectivity, it suffices to show that if v and w are the unique toral semistable vertices of X that map to e, which implies that there exists an edge e of the dual graph associated to X s that abuts to v and w, then ρ b,v = ρ b,w . To this end, we recall that X and X admit natural log structures [determined by the respective multiplicative monoids of regular functions invertible outside the respective special fibers X s , X s ] such that the morphism X X extends uniquely to a morphism of log schemes [cf., e.g., the subsection in Notations and Conventions entitled “Log schemes”; the discussion  e of the local ring of of [Hur], §3.7, §3.8, §3.10]. Moreover, the completion O X at e is isomorphic to O K [[x , y ]]/(x y a ), where a O K , and x , y denote indeterminates which may be chosen in such a way that the homo-  e O  e induced by X X maps morphism of topological O K -algebras O × N M  x → (x ) · ) · u, for some unit u O e , some uniformizer π O K , some positive integer N , and some nonnegative integer M . [Indeed, the fact that N is necessarily positive follows immediately, in light of our assumptions on v and w, from well-known considerations in intersection theory on the Q-factorial normal schemes X and X .] Then the desired inequality ρ b,v = ρ b,w follows immediately from the fact that, up to a possible permutation of the labels “v” and “w”, it holds that x is invertible at the generic point of [the irreducible component corresponding to] v, but non-invertible at the generic point of [the irreducible component corresponding to] w. 29 Remark 2.1.6. In the notation of Remark 2.1.3, suppose further that Y has split reduction, and that the morphism Y X is an isomorphism. Let X be a toral compactified semistable model relative to X , Y X a morphism over X . Then observe that it follows immediately from Remark 2.1.3, together with Zariski’s Main Theorem, that each normal irreducible component of Y s that maps to a closed point of X s , is of genus 0, and has precisely 1 node maps to a closed point of X s . In particular, it follows immediately from an iterated application of the above observation, together with the theory of pointed stable curves, as exposed in [Knud] i.e., by adding finitely many suitably positioned cusps and then considering the various contraction morphisms that arise from eliminating cusps that there exists a unique, up to unique isomorphism, toral compactified semistable model Y relative to X , together with a uniquely deter- mined morphism Y Y of compactified semistable models over X , such that the following universal property is satisfied: if X is a toral compactified semistable model relative to X such that the morphism Y X admits a factorization Y X X , then the morphism Y X admits a unique factorization Y Y X . That is to say, Y may be thought of as a sort of “universal toralization [over X ]” of Y. In particular, it follows immediately from the existence of universal toralizations, together with the theory of pointed stable curves, as exposed in [Knud] i.e., by adding finitely many suitably positioned cusps and then con- sidering the various contraction morphisms that arise from eliminating cusps that the toral compactified semistable models relative to X form a directed inverse system. Definition 2.2. Let Σ Primes be a nonempty subset; K a mixed character- istic complete discrete valuation field of residue characteristic p; X a hyperbolic curve over K. Write Ω for the p-adic completion of [some fixed] K. Then: (i) Let v be a valuation on a field F that contains K. Write O v for the ring of integers determined by v; m v O v for the maximal ideal of O v . Then we shall say that v is a p-valuation [over K] if O K = O v K [which implies that p m v ]. Here, the phrase “over K” will be omitted in situations where the base field K is fixed throughout the discussion. We shall say that v is primitive if it is a p-valuation such that the only prime ideal of O v that contains p is m v . We shall say that v is residue-transcendental def if it is a p-valuation whose residue field k v = O v /m v is a transcendental extension of the residue field of K. 30  is a universal (ii) In the situation of (i), suppose that v is a p-valuation, that X geometrically pro-Σ covering of X, and that F is a subfield [in a fashion  Ω ) compatible with the given inclusion K → F ] of the function field K( X  of X Ω . Then we shall say that v is point-theoretic if it arises from some  point X(Ω), i.e., if O v F is equal to the subring of F consisting  Ω that are regular of elements F that determine rational functions on X at x̃, and whose value at is contained in O Ω Ω. Thus, every point  X(Ω) determines a corresponding point-theoretic valuation of F . (iii) In the situation of (ii), let Z X be a connected finite étale covering  Z X. Let Z be a compactified equipped with a factorization X semistable model of Z with split reduction. Then we shall write VE(Z) for the finite set [equipped with the discrete topology] of vertices and edges of the dual graph associated to Z s . Note that there is a notion of specialization/generization among elements of VE(Z), i.e., we shall say that a vertex specializes to a node, or, alternatively, that a node generizes to a vertex, if the node abuts to the vertex; a vertex specializes/generizes to a vertex if the two vertices coincide; an edge specializes/generizes to an edge if the two edges coincide. For c 1 , c 2 VE(Z), if c 1 specializes to c 2 , or, equivalently, c 2 generizes to c 1 , then we shall write c 1  c 2 . By allowing Z and Z to vary, we thus obtain a topological space  = lim VE(Z), VE( X) ←− def Z where the transition maps in the inverse limit are induced by the corre- sponding scheme-theoretic morphisms of compactified semistable models [which form a directed inverse system cf. Proposition 2.3, (iii), below], that is to say, by mapping a vertex [i.e., irreducible component] or edge [i.e., node] to the smallest vertex [i.e., irreducible component] or edge [i.e., node] that contains its scheme-theoretic image. [Here, we recall that any such morphism of compactified semistable models always maps a smooth closed point that is isolated in the fiber of the morphism to a smooth closed  as a VE- point cf. Remark 2.1.3.] We shall refer to an element of VE( X)  chain of X. Note that the notion of specialization/generization among el- ements of each VE(Z) determines [i.e., by considering each constituent set in the above inverse limit] a notion of specialization/generization among  We shall say that an element c VE( X)  is primitive elements of VE( X). if every generization of c is equal to c. 31 (iv) In the situation of (iii), let z c 1 , z c 2 VE(Z). Then we shall write δ(z c 1 , z c 2 ) Z for the integer δ such that the set of vertices contained in a path of minimal length between z c 1 and z c 2 on the dual graph of Z s is of cardinality δ + 1.  Then we shall write Let c 1 , c 2 VE( X). def δ(c 1 , c 2 ) = sup δ(w c 1 , w c 2 ) Z {+∞}, Z where Z ranges over the set of compactified semistable models with split reduction of connected finite étale coverings Z X equipped with a  Z X; w c VE(Z) denotes the element determined factorization X i by c i for each i = 1, 2. (v) In the situation of (ii), suppose further that F contains the function field  of X.  Then observe that v determines, by considering the centers K( X) associated to v on the various “Z” in the discussion of (iii), an element  which we shall refer to as the center-chain associated to v. VE( X),  In particular, any point X(Ω) determines, by considering the point-  which we shall theoretic valuation associated to x̃, an element VE( X), refer to as the [point-theoretic] center-chain associated to x̃. Write x  X(Ω) for the image of in X(Ω). Thus, the Gal( X/X K )-orbit of is  completely determined by x. We shall refer to the Gal( X/X K )-orbit of the center-chain associated to as the [point-theoretic] orbit-center-chain associated to x [cf. the discussion of Remark 2.2.4 below]. (vi) In the situation of (ii), let Z X be a connected finite étale covering  Z X. Let Z be a compactified equipped with a factorization X semistable model with split reduction of Z. In the remainder of the dis- cussion of the present item (vi), all toral compactified semistable models relative to Z will be assumed to have generic fibers that are equipped  Z. Write with the structure of a subcovering of the pro-covering X V(Z) (respectively, E(Z))  for the set of vertices (respectively, edges) of Z s . Thus, VE(Z) = V(Z) E(Z). For each c VE(Z), write V c for the set of equivalence classes of the set of vertices of toral compactified semistable models relative to Z that map to the closed subscheme of Z s corresponding to c, where we apply the equivalence relation induced by the dominant morphisms over Z of toral compactified semistable models relative to Z. Let e E(Z). Then we shall write V e for the union of V e and the vertices of V(Z) that abut to e. Observe that, for each toral compactified semistable model Z relative to Z and 32 each c VE(Z ) that maps to the closed subscheme of Z s corresponding to c, the set V c may be regarded, in a natural way [i.e., by considering the maps induced by dominant morphisms over Z of toral compactified semistable models relative to Z], as a subset of V c . We shall refer to such a subset V c V c as a basic open subset of V c . Thus, from the point of view of the natural bijection, determined by selecting a branch b of the edge e, between V e and the set of rational numbers ρ such that 0 < ρ < 1 [cf. Remarks 2.1.4, 2.1.5, 2.1.6], the basic open subsets V e correspond precisely to the open intervals with rational endpoints of V e . In particular, it is natural to regard V c as being equipped with the topology determined by the open basis consisting of the basic open subsets V c . We shall refer to as a quasi-basic open subset of V e any open subset of V e which is a union of a countable collection of basic open subsets V e for which the relation of inclusion determines a total ordering. We shall refer to as a Dedekind cut of V e an unordered pair {D 1 , D 2 } of disjoint nonempty quasi-basic open subsets D 1 , D 2 V e such that V e = D 1 D 2 . Write D e for the set of Dedekind cuts of V e . Note that the topology of the V w , where w V e \ V e , induces, in a natural way, a topology on the set  def T e = V e D e [i.e., by taking as an open basis for the topology for T e the subsets of T e obtained as the intersections with T e of unions of an open subset U V w , where w V e \ V e , with the set of Dedekind cuts {D 1 , D 2 } D e such that both D 1 and D 2 intersect U ]. Thus,  def T e = V e D e . [with the induced topology] is homeomorphic to the open interval (0, 1) R of the real line [cf. Remarks 2.1.4, 2.1.5, 2.1.6]. Write def VE(Z) tor = V(Z)  T e . e∈E(Z) Thus, the discrete topology on V(Z), together with the topologies defined above on the T e , determine a topology on VE(Z) tor . Moreover, there exists a noncontinuous [cf. Remark 2.2.1 below] natural surjective map  Z : VE(Z) tor −→ VE(Z) that maps each T e to e. Finally, by allowing Z and Z to vary, we thus obtain a topological space  tor = lim VE(Z) tor VE( X) ←− def Z [cf. the discussion of (iii)], together with a natural [not necessarily surjec- tive!] map  tor −→ VE( X).   X  : VE( X) 33 (vii) We shall say that X satisfies Σ-RNS [i.e., “Σ-resolution of nonsingulari- ties” cf. [Lpg1], Definition 2.1] if the following condition holds: Let v be a discrete residue-transcendental p-valuation on the function field K(X) of X. Then there exists a connected geo- metrically pro-Σ finite étale Galois covering Y X such that Y has stable reduction [over its base field], and v coincides with the restriction [to K(X)] of a discrete valuation on the function field K(Y ) of Y that arises from an irreducible component of the special fiber of the stable model of Y . Remark 2.2.1. In the notation of Definition 2.2, (vi), the natural surjective map  Z is not continuous in general. Indeed, to see this, it suffices to observe that the inverse image of the closed subset consisting of a single edge is an open subset of VE tor (Z) that is not closed. Finally, we observe that one may also conclude from this noncontinuity of  Z that  X  is not continuous. Remark 2.2.2. In the notation of Definition 2.2, (vii), suppose that Σ \ {p} is nonempty, and that X satisfies Σ-RNS. Then, by considering a suitable admis- sible covering of the stable model of “Y as in Definition 2.2, (vii), one verifies immediately that one may assume that the normalization of the irreducible component that appears in Definition 2.2, (vii), is of genus 2. Remark 2.2.3. In the notation of Definition 2.2, (vii), we make the following observations. (i) Let L Ω be a topological subfield containing K that arises as the [topo- logical] field of fractions of a mixed characteristic complete discrete valu- ation field of residue characteristic p. Then let us observe that any compactified semistable model of X L over O L arises, after possibly replacing K and L, respectively, by suitable finite ex- tension fields of K and L, as the result of base-changing, from O K to O L , some compactified semistable model of X K over O K . Indeed, since every element of L admits arbitrarily close p-adic approxi- mations by elements of finite extension fields of K contained in K, this observation follows immediately by noting that it follows immediately from the well-known theory of pointed stable curves, as exposed in [Knud], that, after possibly replacing K and L, respectively, by suitable finite extension fields of K and L and possibly replacing X by some dense open subscheme of X, we may assume without loss of generality that the given compactified semistable model of X L over O L is in fact the [unique, up to unique iso- morphism] compactified stable model of X L , i.e., which necessarily arises as the result of base-changing, from O K to O L , the [unique, up to unique isomorphism] compactified stable model of X K over O K . 34 (ii) We maintain the notation of (i). Then let us observe that X satisfies Σ-RNS if and only if X L satisfies Σ-RNS. Indeed, this observation follows immediately, in light of the observation of (i), from Proposition 2.3, (ii), (iii), below; Proposition 2.4, (iv), below [cf. also Definition 2.2, (vii), as well as the discussion of the final portion of the subsection in Notations and Conventions entitled “Fundamental groups”]. (iii) Next, let L be a mixed characteristic complete discrete valuation field of residue characteristic p that contains K as a topological subfield. Then observe that it follows immediately from the well-known elementary theory of complete discrete valuation fields that L is isomorphic, as a topological K-algebra, to a field “L” of the sort discussed in (i), (ii) if and only if [the valuation on] L is not residue-transcendental [relative to K], i.e., if and only if the residue field of L is an algebraic extension of the residue field of K. (iv) We maintain the notation of (iii). Then let us observe that if X L satisfies Σ-RNS, then the residue field of L is an algebraic extension of the residue field of K. Indeed, it suffices to verify this observation after replacing K by a finite extension field of K. In particular, we may assume without loss of gener- ality [cf. Proposition 2.3, (iii)] that there exists a compactified semistable model X of X over O K . Then it follows immediately from the unique- ness, up to unique isomorphism, of compactified stable models [cf. also Definition 2.2, (vii), as well as the discussion of the final portion of the subsection in Notations and Conventions entitled “Fundamental groups”; the stable reduction theorem of [DM], [Knud]], that if X L satisfies Σ-RNS, then any closed point of (X O K O L ) s that arises as the center of a dis- crete residue-transcendental p-valuation on the function field of X L is necessarily defined over some algebraic extension of the residue field of K. On the other hand, this contradicts the existence of discrete residue- transcendental p-valuations on the function field of X L that arise as the local rings of generic points of exceptional divisors of blow-ups of smooth closed points of (X O K O L ) s that are not defined over some algebraic extension of the residue field of K. This completes the proof of the above observation. (v) We maintain the notation of (iii). Then we observe further that, under the assumption that X satisfies Σ-RNS, it holds that the residue field of L is an algebraic extension of the residue field of K if and only if X L satisfies Σ-RNS. 35 Indeed, necessity follows formally from the observations of (ii), (iii), while sufficiency follows formally from the observation of (iv). Remark 2.2.4. In the context of Definition 2.2, we recall from the general theory of valuations the following well-known basic facts. Let L be a field equipped with a valuation v, M a finite normal extension field of L. Write O v for the ring of integers of L with respect to v, m v O v for the maximal ideal of O v , O M for the integral closure of O v in M , v M/L for the set of valuations on M that extend v, and Aut(M/L) for the group of automorphisms of M that restrict to the identity on L. If w v M/L , then we shall write O w for the ring of integers of def M with respect to w, m w O w for the maximal ideal of O w , p w = O M m w . Then the set v M/L is nonempty [cf. [EP], Theorem 3.1.1], and the natural action of Aut(M/L) on v M/L is transitive [cf. [EP], Theorem 3.2.14]. Moreover, O M  = O w w∈v M/L [cf. [EP], Theorem 3.1.3, (2)]; the assignment v M/L  w → p w determines a bijective correspondence between v M/L and the set of prime ideals of O M that lie over m v , and, for w v M/L , O w = (O M ) p w [cf. [EP], Theorem 3.1.1; [EP], Theorem 3.2.13]. In this situation, if we assume further that v is real, and that L is complete with respect to v, then v M/L is of cardinality 1 [cf. [Neu], Chapter II, Theorem 4.8]. [Here, we recall that if v is not real, then O v does not, in general, satisfy Hensel’s Lemma, i.e., even if L is complete with respect to v [cf. [EP], Remark 2.4.6].] More generally, if v is real, then L admits a  [cf. [EP], Theorem 1.1.4], which is a henselian field [cf. natural completion L [Neu], Chapter II, Theorem 4.8; the discussion preceding [EP], Lemma 4.1.1] and contains, up to natural isomorphism, the henselization L h of L [cf. [EP], the discussion preceding Theorem 5.2.2] as a subfield, i.e.,  L h L [cf. [EP], Corollary 4.1.5; [EP], Corollary 5.2.3; the discussion of Case 2 in the proof of [EP], Theorem 6.3.1]. The various basic properties stated in the following Proposition 2.3 consist of elementary results that are essentially well-known or implicit in the literature [cf. Remarks 2.3.2, 2.3.3 below], but we give [essentially] self-contained state- ments and proofs here in the language of the present discussion for the sake of completeness. Proposition 2.3 (Basic properties of models of hyperbolic curves). Let K be a mixed characteristic complete discrete valuation field of residue charac- teristic p; X a hyperbolic curve over K. Write K(X) for the function field of X. Then the following hold: 36 (i) Let R K(X) be a finitely generated normal O K -subalgebra whose field of fractions coincides with K(X). Then Spec R arises as an open subscheme of a compactified model of X over O K . (ii) Let v be a discrete residue-transcendental [cf. Remark 2.3.1 below] p- valuation on K(X). Then v arises as the discrete valuation associated to an irreducible component of the special fiber of a compactified model of X over O K . (iii) Let X be a compactified model of X over O K equipped with the action of a finite group G by O K -linear automorphisms [which thus restrict to K- linear automorphisms of X]. Then, after possibly replacing K by a finite field extension of K, there exists a compactified semistable model of X over O K that dominates X and is stabilized by the action of G on X. (iv) Let Y X be a [connected] finite étale Galois covering of hyperbolic curves over K, Y sst a compactified semistable model of Y over O K that def is stabilized by G = Gal(Y /X). Write X for the quotient of Y sst by the natural action of G on Y sst . Then X is a compactified semistable model of X over O K , and the images of smooth points of Y s sst via the natural morphism Y s sst X s are smooth points of X s . Moreover, the image of a node of Y s sst via the natural morphism Y s sst X s is a node of X s if and only if G does not permute the branches of the node. In particular, the dual graph of X s may be reconstructed from the dual graph of Y s sst , together with the action of G on the dual graph of Y s sst . Finally, this reconstruction procedure is functorial, with respect to maps of vertices/edges to vertices/edges [i.e., as in the discussion of Defi- nition 2.2, (iii)], on the category of [connected] finite étale Galois coverings of X over K. (v) Suppose that we are in the situation of Definition 2.2, (ii), (iii), (iv), (v).  to its associated Then the assignment that maps a p-valuation on K( X) center-chain determines bijections as follows:  p-valuations on K( X)  VE( X),  primitive p-valuations on K( X)  prim , VE( X)  pt-th ,  VE( X) X(Ω)  pt-th /Gal( X/X  X(Ω) VE( X) K ),  prim VE( X)  denotes the subset of primitive VE-chains, where VE( X) pt-th   and VE( X) VE( X) denotes the subset of point-theoretic center- chains. 37  Write R c (vi) Suppose that we are in the situation of (v). Let c VE( X).  K( X) for the valuation ring of the p-valuation associated to c [cf. (v)]. Then it holds that R c = lim O Z,z c , −→ Z where the direct limit ranges over the set of compactified semistable models with split reduction Z of the domain curves of connected finite étale cov-  Z X; z c denotes the erings Z X equipped with a factorization X center on Z determined by R c ; the transition maps in the direct limit are induced by the corresponding scheme-theoretic morphisms of compactified semistable models [cf. the discussion of Definition 2.2, (iii)]. (vii) Suppose that we are in the situation of (v). Suppose, moreover, that X is  is primitive if and only if it is either proper. Then a p-valuation of K( X) real or point-theoretic. Equivalently,  consists of the disjoint the set of primitive p-valuations of K( X)  and the union of the non-point-theoretic real p-valuations of K( X)  point-theoretic p-valuations of K( X).  an for the topological pro-Berkovich space as- In particular, if we write X sociated to [i.e., the inverse limit of the underlying topological spaces of  then the Berkovich spaces associated to the finite subcoverings of ] X,  may be naturally identified the set of primitive p-valuations of K( X) an  with the underlying set of X . (viii) Suppose that we are in the situation of (vii). Then there exists a natural commutative diagram of maps of sets  an X    tor −−−−→ VE( X) θ X    X  prim −−−−→ VE( X),  VE( X) ι X  where the upper horizontal arrow θ X  is a homeomorphism [cf. Remark 2.3.3 below]; the lower horizontal arrow ι X  denotes the natural inclu- sion; the left-hand vertical arrow denotes the bijection obtained by form- ing the composite of the natural identification that appears in the state- ment of (vi) with the second bijection in the display of (v); the right- hand vertical arrow  X  denotes the natural morphism [cf. Definition 2.2, (vi)]. In particular,  X  is injective and in fact admits a natural split-  VE( X)  tor [i.e., such that τ    is the identity on ting τ X  : VE( X) X X  tor ]. On the other hand, neither ι  nor   is surjective [cf. Remark VE( X) X X 2.3.4 below]. 38  be distinct (ix) Suppose that we are in the situation of (v). Let c 1 , c 2 VE( X) elements. Then one of the following conditions holds: δ(c 1 , c 2 ) = +∞.  such δ(c 1 , c 2 ) = 0, and there exists a unique element c 3 VE( X) that c 3  c 1 and c 3  c 2 . In particular, if c 1 and c 2 are distinct primitive elements, then it holds that δ(c 1 , c 2 ) = +∞.  Then the (x) Suppose that we are in the situation of (v). Let c VE( X). cardinality of the set  \ {c} | c   c} {c  VE( X) is at most 1. (xi) Suppose that we are in the situation of (v). Let Σ Primes be a subset; l Σ \ {p}; H G K a closed subgroup such that the restriction to H of the l-adic cyclotomic character of K has open image, and, moreover, the intersection H I K of H with the inertia subgroup I K of G K admits (Σ) a surjection to [the profinite group] Z l ; s : H Π X a section of the (Σ) restriction to H of the natural surjection Π X  G K . Then there exists  prim that is fixed by the restriction, via s, to H of an element c VE( X) (Σ)  prim VE( X).  In particular, if X is the natural action of Π X on VE( X) an an  proper, then there exists an element c X [cf. (vii)] that is fixed by (Σ) the restriction, via s, to H of the natural action of Π X on the topological  an . pro-Berkovich space X (xii) Let Σ Primes be a subset of cardinality 2 that contains p. Then there exists a connected geometrically pro-Σ finite étale Galois covering X X satisfying the following conditions: X has split stable reduction. Write X for the [unique, up to unique isomorphism] stable model of X . Then X s is singular, and every irreducible component of X s is a smooth curve of genus 2. Proof. First, we verify assertion (i). Since R is a finitely generated algebra over O K (⊆ R), it follows that Spec R admits an embedding over O K into N N -dimensional affine space A N O K for some positive integer N . Write Z P O K N N for the scheme-theoretic closure of the image of Spec R in P O K (⊇ A O K ); Z for the normalization of Z . Thus, the structure sheaf O Z is p-torsion-free, hence flat over O K . Since, moreover, Z is [of finite type over the complete discrete valuation ring O K , hence] excellent, it follows that Z is a proper, flat scheme of finite type over O K , whose generic fiber may be identified with the [uniquely 39 determined, up to unique isomorphism] smooth compactification of X over K. This completes the proof of assertion (i). Next, we verify assertion (ii). Write A K(X) for the discrete valuation ring associated to v. Note that A may be written as the direct limit [i.e., in fact, union] of a direct system of finitely generated subalgebras {A i A} i∈I over O K . Moreover, since the field extension K K(X) is finitely generated, and each A i is [finitely generated over the complete discrete valuation ring O K , hence] excellent, we may assume without loss of generality, i.e., by replacing A i by its normalization in K(X), that each A i is normal with field of fractions equal to K(X). Write p i A i for the prime ideal determined by the maximal ideal of A. Thus, since v is a p-valuation [over K], it follows immediately that each of the natural inclusions O K → (A i ) p i → A is a homomorphism of local rings. Next, let us observe that since the residue field extension determined by the natural inclusion O K A of local rings is assumed to be transcendental, it follows that there exists an element i I such that the residue field k(p i ) of p i is a transcendental extension of the residue field of O K . Let Z be a compactified model of X over O K that contains Spec A i as an open subscheme [cf. Proposition 2.3, (i)]. Then since Z is of dimension 2 [cf. Remark 2.1.1], it follows that the height of p i is equal to 1 or 2. On the other hand, if p i is of height 2, then it follows that p i corresponds to a closed point of Z s , hence that k(p i ) is a finite extension of the residue field of O K , i.e., in contradiction to our assumption of transcendality. Thus, we conclude that p i is of height 1, hence that (A i ) p i is a discrete valuation ring whose field of fractions is equal to K(X). But this implies that (A i ) p i = A. This completes the proof of assertion (ii). Next, we verify assertion (iii). First, we observe that, after possibly replac- ing K by a suitable finite extension field of K, there exists a G-equivariant finite morphism X P 1 O K to the projective line over O K [equipped with the trivial action by G]. Indeed, since O K is a complete discrete valuation ring, by deforming any [suitably large positive power of a] very ample line bundle on the projective curve X s , we obtain a very ample line bundle L on X , hence, after possibly replacing K by a suitable finite extension field of K, a pair of global sections σ 1 , σ 2 of the line bundle L such that the G-orbit of the zero locus of σ 1 is disjoint from the G-orbit of the zero locus of σ 2 . Thus, for i = 1, 2, the product σ i G of the G-translates of σ i determines a global section of the [still very ample!] tensor product L G of G-translates of L such that σ 1 G and σ 2 G still have disjoint zero loci, hence determine a G-equivariant finite morphism X P 1 O K over O K , as desired. Fix such a finite morphism, and write f K(X) for the rational function on X determined by the standard coordinate function on P 1 O K . Next, we recall that it follows from the stable reduction theorem [cf. [DM], [Knud]] that, after possibly replacing K by a suitable finite extension field of K, we may assume without loss of generality that every closed point in the support Supp(f ) [in the smooth compactification of X] of the principal divisor associ- ated to f is K-rational, and that X has stable reduction over K. Moreover, by replacing X by a suitable G-stable open subscheme of X, we may assume with- out loss of generality that Supp(f ) is contained in the set of cusps of X. Write X for the compactified stable model of X over O K ; E X for the reduced 40 closed subscheme determined by the set of closed points where an irreducible component of the zero divisor of f on X intersects an irreducible component of the divisor of poles of f on X . Thus, the action of G on X extends to X .  X ,x for the completion of the local ring of X at x. Fix a Let x E. Write O uniformizer π K O K . Next, suppose that x is a smooth closed point of X . Then there exist nonzero integers a, b of opposite sign and a unit u (O K [[t]]) × [where t denotes an indeterminate], together with an isomorphism of topological O K -algebras  X ,x (  X ,x O K [[t]], such that the image of f in the field of fractions of O O b O K [[t]]) is of the form u·t a ·π K . Next, observe that, by replacing K by a suitable finite extension field of K [so it may no longer be the case that the element “π K is a uniformizer of O K !], we may assume without loss of generality that there −b exists an element γ O K such that γ a = π K . Write x η for the K-valued point of the smooth compactification of X determined by the section of the structure morphism X Spec O K corresponding to the homomorphism of topological O K -algebras  X ,x →) ( O O K [[t]] −→ O K that maps t → γ O K ; x  η for the K-valued point of the smooth compactifica- tion of X determined by the section of the structure morphism X Spec O K corresponding to the homomorphism of topological O K -algebras  X ,x →) O K [[t]] −→ O K ( O that maps t → 0 O K ; X [x] for the compactified stable model of X \ {x η , x  η } over O K . Thus, it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that the natural inclusion X \ {x η , x  η } → X determines a natural birational, dominant morphism X [x] X . Finally, we observe that it follows immediately from the various definitions involved that the rational function f is a unit [at x η , hence] at the generic point of the unique irreducible component of (X [x]) s that maps to a closed point of X s ; in particular, the zero divisor of f does not intersect the divisor of poles of f in some Zariski neighborhood of this irreducible component. Next, suppose that x is a nonsmooth closed point of X . Then since the  X ,x ) s are Q-Cartier divisors, it follows two irreducible components of (Spec O r that there exist positive integers a, b, r and a unit u (O K [[s, t]]/(st π K )) × [where s, t denote indeterminates], together with an isomorphism of topological r  X ,x O K -algebras O O K [[s, t]]/(st−π K ), such that the image of some positive r  power of f in the field of fractions of O X ,x ( O K [[s, t]]/(st π K )) is of the a −b form u·s ·t . Next, observe that, by replacing K by a suitable finite extension field of K [so it may no longer be the case that the element “π K is a uniformizer of O K !], we may assume without loss of generality that there exists an element γ O K such that γ a+b = π K . Write x η for the K-valued point of the smooth compactification of X corresponding to the section of the structure morphism X Spec O K induced by the homomorphism of topological O K -algebras  X ,x →) O K [[s, t]]/(st π r ) −→ O K ( O K 41 def that maps s → γ br O K , t → γ ar O K ; x  η = x η ; X [x] for the compactified stable model of X \ {x η , x  η } = X \ {x η } over O K . Thus, it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that the natural inclusion X \ {x η , x  η } = X \ {x η } → X determines a natural birational, dom- inant morphism X [x] X . Finally, we observe that it follows immediately from the various definitions involved that the rational function f is a unit [at x η , hence] at the generic point of the unique irreducible component of (X [x]) s that maps to a closed point of X s ; in particular, the zero divisor of f does not in- tersect the divisor of poles of f in some Zariski neighborhood of this irreducible component. Next, observe that the underlying set of E is finite. Thus, by replacing X by a suitable G-stable open subscheme of X, we may assume without loss of generality that for each x E, the K-valued points x η , x  η constructed above are contained in the set of cusps of X. Then it follows immediately from the above discussion that the zero divisor of f on X does not intersect the divisor of poles of f on X . But this implies that f determines a G-equivariant domi- nant morphism X P 1 O K over O K whose restriction to the respective generic fibers coincides with the restriction to the respective generic fibers of the finite morphism X P 1 O K over O K constructed above. Thus, since X is normal, we conclude that the morphism X P 1 O K admits a factorization X X P 1 O K , as desired. This completes the proof of assertion (iii). Next, we verify assertion (iv). First, we observe that since the operation of forming the quotient of Y sst by G commutes with flat base-change, one verifies immediately that it suffices to verify assertion (iv) after performing any finite, faithfully flat base-change from O K to the ring of integers in a finite extension field of K. In particular, by replacing K by a suitable finite extension field of K, we may assume without loss of generality that the cusps of X are K-rational, and that X has stable reduction over K [cf. [DM], [Knud]]. [In fact, these conditions are satisfied even if one does not pass to a finite extension field of the original given K, but we omit a proof of this fact since it is not logically necessary for the present discussion.] Write X st for the compactified stable model of X over O K . Next, let us observe that, after possibly replacing K by a suitable finite extension field of K, one may regard Y sst as the compactified stable model associated to the hyperbolic curve Y  obtained by removing from Y a collection of G-orbits of K-rational points of Y such that the cardinality of the set of G-orbits of closed points of Y s sst contained in the intersection of any irreducible component of Y s sst with the image of the corresponding collection of O K -rational points of Y sst is 3. In particular, one verifies immediately that, by replacing Y by Y  , we may assume without loss of generality that the cardinality of the set of closed points of each irreducible component of X s that lie in the image of the cusps of Y sst is 3. Next, let us observe that it follows immediately from the definition of the natural quotient morphism Y sst X that the natural morphism Y sst X st over O K induced by the morphism Y X [cf. [ExtFam], Theorem A] admits 42 a factorization Y sst −→ X −→ X st , where we note that it follows immediately from the definition of X that X is a compactified model of X over O K . Thus, it follows immediately from the above discussion of Y sst , together with the theory of pointed stable curves, as exposed in [Knud], that the morphism X X st is birational and quasi-finite, hence, by Zariski’s Main Theorem, an isomorphism. In particular, we conclude that X is a compactified semistable model of X over O K . Moreover, it follows immediately from Remark 2.1.3 that the natural morphism Y s sst X s maps smooth points of Y s sst to smooth points of X s . Next, let e Y Y s sst be a node. Write e X X s for the image of e Y via the natural morphism Y s sst X s ; G e Y G for the stabilizer of e Y in G; Y e sst for the Y spectrum of the completion of the local ring of Y sst at e Y ; X e X for the spectrum of the completion of the local ring of X at e X . Thus, G e Y acts naturally on Y e sst ; X e X may be identified with the quotient of Y e sst by the action of G e Y ; the Y Y set B Y of irrreducible components of (Y e sst ) may be identified with the set [of Y s cardinality 2] of branches of e Y ; the set B X of irrreducible components of (X e X ) s is of cardinality 1 if and only if e X is a smooth point of X s and may be identified with the set [of cardinality 2] of branches of e X whenever e X is a node. On the other hand, it follows immediately from elementary commutative algebra that the set B X may be naturally identified with the set of G e Y -orbits of B Y . The remaining portion of assertion (iv) now follows formally. This completes the proof of assertion (iv).  Write Next, we verify assertions (v) and (vi). Let c VE( X). def R c = lim O Z,z c , −→ Z where Z ranges over the compactified semistable models with split reduction of the domain curves of connected finite étale coverings Z X equipped  Z X; with a factorization X c Z denotes the irreducible component or node of Z s determined by c; z c denotes the generic point of the intersection of the [closed irreducible] images in Z s of the c Z associated to compactified semistable models with split reduction of domain curves of connected finite étale coverings  Z Z X such that the Z X equipped with a factorization X morphism Z Z extends to a morphism Z Z; the transition maps in the direct limit are the homomorphisms of local rings induced by the corresponding scheme-theoretic morphisms of com- pactified semistable models [which form a directed inverse system cf. Proposition 2.3, (iii)]. 43 Then it follows immediately from the various definitions involved that the field  that O K R c , and that R c is a local of fractions of R c coincides with K( X),  c K( X)  be a valuation domain whose maximal ideal m R c contains p. Let R  c for the ring that dominates R c [cf., e.g., [EP], Theorem 3.1.1]. Write m R  c R  c . Thus, since O K O R c R  c and p m R m  , we maximal ideal of R c K R c  c , i.e., that the valuation determined by the valuation conclude that O K = K R  c is a p-valuation. ring R  c may be written as the direct limit [i.e., in fact, union] of a direct Note that R  c } i∈I over O K . Moreover, system of finitely generated subalgebras {R i R since R i is [finitely generated over the complete discrete valuation ring O K ,  hence] excellent, by replacing R i by its normalization in the subfield of K( X) generated by the field of fractions of R i and some suitable finite extension field of K, we may assume without loss of generality that R i is normal, and that there exists a compactified semistable model with split reduction Z i of the domain curve of a connected finite étale covering Z i X equipped with a  Z i X such that Spec R i arises as an open subscheme of factorization X def Z i [cf. Proposition 2.3, (i), (iii)]. Write p i = m R  c R i R i . Thus, if we write z i for the point of Z i that corresponds to p i , then it holds that  c = lim O Z ,z . R i i −→ i∈I On the other hand, observe that, since m R  c R c = m R c , it follows immediately from the various definitions involved that each “O Z i ,z i (= (R i ) p i )” of the above direct limit appears as one of the “O Z,z c in the direct limit used to define R c .  c , hence that R c = R  c , i.e., that R c  c R c R In particular, we conclude that R is the valuation ring associated to a p-valuation. Thus, in summary, we obtain a natural map  −→ p-valuations on K( X)  VE( X)   c → R c . Moreover, one verifies immediately that this map that maps VE( X) defines an inverse to the natural map  p-valuations on K( X)  −→ VE( X) in the statement of assertion (v), hence that both of these maps are bijective. Since both of these maps are manifestly compatible with specialization/generization, we thus conclude that these induce a bijection  primitive p-valuations on K( X)  prim VE( X) as in the statement of assertion (v). Thus, to complete the proof of assertion (v), it suffices to verify that the natural map   X(Ω) −→ p-valuations on K( X) 44  [i.e., that assigns to an element of X(Ω) the associated point-theoretic valuation   on K( X)] is injective. To this end, let X(Ω). Write v for the point-theoretic  valuation on K( X) associated to x̃. Then observe that is defined over K if and  is strict. If this inclusion is strict, then the only if the inclusion O v · K K( X)  is the valuation ring determined [i.e., in the usual sense subring O v · K K( X) of the classical theory of one-dimensional function fields over algebraically closed  is strict, the fields] by x̃. In particular, whenever the inclusion O v · K K( X)   point X(K) (⊆ X(Ω)) is completely determined by v. Thus, it remains to  In this case, the valuation v is real, and consider the case where O v ·K = K( X).  induces, by passing to the respective completions, an the inclusion K K( X)  with respect to v. In particular, isomorphism of Ω with the completion of K( X)  we obtain a natural homomorphism K( X) Ω, which completely determines  the point X(Ω). This completes the proof of assertion (v). Assertion (vi) follows immediately from the proof of assertion (v).  Next, we verify assertion (vii). Let x X(Ω). Write v for the point-theoretic  valuation on K( X) associated to x; φ x : O v O Ω for the homomorphism obtained by evaluating rational functions at x. Let q O v be a prime ideal that contains p. Then observe that it follows immediately from the construction of v that p 1 ·Ker(φ x ) Ker(φ x ). Since p q, we thus conclude that Ker(φ x ) q. On the other hand, observe that [it follows immediately from the construction of v that] this inclusion implies that q contains [hence coincides with] the radical of the ideal (p, Ker(φ x )) O v , which is easily seen to be equal to the maximal ideal m v of O v . Thus, we conclude that v is primitive.  Let a m v . Then since v is real, Next, let v be a real p-valuation on K( X). there exists a positive integer N such that a N (p). In particular, any prime ideal that contains p contains [hence coincides with] m v . Thus, we conclude that v is primitive.  For each z O v , write Next, let v be a primitive p-valuation on K( X).  for the O v -subalgebra generated by 1 . Thus, if (K ⊆) (O v ) p = (O v ) z K( X) z  then it follows immediately from the classical theory of one-dimensional K( X), function fields over algebraically closed fields that v is a point-theoretic valuation.  Note Therefore, we may assume without loss of generality that (O v ) p = K( X). that this implies that for each x O v \ {0}, there exist a positive integer N and y O v such that p N = xy. Moreover, in this situation, it holds that m K O v = m v . Indeed, since v is a p-valuation, the inclusion m K O v m v is immediate. Now suppose that there exists an element x m v \ m K O v . Then it follows that 1 p (O v ) x , hence that there exists a prime ideal p v of O v such that x p v , and p p v . On the other hand, since v is primitive, we conclude that x m v = p v , a contradiction. This completes the proof of the equality in the above display. Note that this equality implies that for each x m v \ {0}, there exist a positive integer N and y O v such that x N = py. In particular, it follows immediately from the various definitions involved that v coincides with the real valuation 45 determined by the assignment O v \ {0}  x → sup{ Q | x p · O v } = inf{ Q | x −1 p · O v } R. This completes the proof of assertion (vii). Next, we verify assertion (viii). First, let us observe that it follows immedi- ately from the final portion of Proposition 2.3, (vii), that the natural map   an −→ VE( X) X  the center-chain associated to the i.e., that assigns to a valuation on K( X) valuation admits a factorization  an  prim −→ VE( X),  X VE( X) ι X  where the first arrow is a bijection, and the second arrow ι X  denotes the natural inclusion. On the other hand, it follows immediately from the discussion of the  tor in Definition ratios “ρ b,v in Remark 2.1.4 and the construction of “VE( X)  also admits a factorization  an VE( X) 2.2, (vi), that this natural map X  tor −→ VE( X),   an −→ VE( X) X θ X   X where the first map θ X  is defined by considering ratios “ρ b,v as in Remark 2.1.4  tor in Definition 2.2, (vi)], and the second [cf. also the construction of “VE( X) arrow  X  is the natural map discussed in the final portion of Definition 2.2, (vi). In particular, we obtain a commutative diagram of maps of sets  an X    tor −−−−→ VE( X) θ X    X  prim −−−−→ VE( X).  VE( X) ι X  Note that the commutativity of the diagram already implies that θ X  is injective. Moreover, one verifies immediately i.e., by considering suitable “v” as in Remark 2.1.4 that each composite map θ Z  tor −→ VE(Z) tor  an −→ VE( X) X θ X  where the second arrow is the natural projection arising from the inverse limit  tor [cf. Definition 2.2, (vi)] has dense image. Thus, in the definition of VE( X)  an [cf. [Brk], Theorem 1.2.1], together it follows from the compactness of X with the easily verified fact [cf. the construction of Definition 2.2, (vi)] that VE(Z) tor is Hausdorff, that to verify that θ X  is a homeomorphism, it suffices to verify that each map θ Z is continuous. Moreover, once one knows that θ X  is a homeomorphism, one may construct a natural splitting τ X  as in the statement 46 of Proposition 2.3, (viii), by constructing a natural splitting of the natural  VE( X)  prim inclusion ι X  . On the other hand, such a natural splitting VE( X) of ι X  is implicit in the content of Proposition 2.2, (x) [which will be verified below, independently of the present assertion (viii)], i.e., one assigns to each  the unique generization VE( X)  prim of c. nonprimitive element c VE( X) Thus, in summary, to complete the proof of assertion (viii), it suffices to verify that each map  an −→ VE(Z) tor θ Z : X as in the above discussion is continuous. Let Z be a toral compactified semistable model relative to Z. Then we shall refer to an open subscheme U of Z s as a componentwise open of Z if U is an open subscheme of Z s whose underlying open subset is the complement of a node or an irreducible component of Z s . Observe that it follows immediately from the construction of VE(Z) tor given in Definition 2.2, (vi), that each componentwise open of each toral compactified semistable model relative to Z determines, in a natural way, a closed subset of VE(Z) tor . We shall refer to the closed subsets of VE(Z) tor obtained in this way as componentwise closed subsets of VE(Z) tor . Note that it follows immediately from the construction of VE(Z) tor given in Definition 2.2, (vi), that the com- plements of the componentwise closed subsets of VE(Z) tor form an open basis of the topology of VE(Z) tor . Thus, to complete the proof of the contininuity of θ Z , it suffices to verify that the inverse image via θ Z of any componentwise  an . But this follows immediately from closed subset of VE(Z) tor is closed in X the definition of the topology of the Berkovich spaces [cf. the discussion of [Brk],  an in the §1.1, §1.2] that appear in the inverse limit that is used to define X statement of Proposition 2.3, (vii). This completes the proof of assertion (viii). Next, we verify assertion (ix). In the following, we assume that δ(c 1 , c 2 ) = +∞. Let us first consider the case where δ(c 1 , c 2 ) 1. Then it follows immediately from the definition of δ(−, −) that there exists a compactified semistable model Z with split reduction of a connected finite étale covering Z X equipped  Z X such that δ(z c , z c ) 1 [cf. the notation with a factorization X 1 2 of Proposition 2.3, (vi)]. In particular, there exists a node e of Z s that does not coincide with z c 1 or z c 2 , and whose corresponding edge lies on a path of minimal length between z c 1 and z c 2 . On the other hand, by considering suitable torally compactified semistable models relative to Z at e, we conclude that δ(c 1 , c 2 ) = +∞, in contradiction to our assumption that δ(c 1 , c 2 ) = +∞. Thus, to complete the proof of assertion (ix), it suffices to consider the case where δ(c 1 , c 2 ) = 0. Let us first observe that the condition that δ(c 1 , c 2 ) = 0  such that c 3  c 1 and c 3  c 2 . implies the existence of an element c 3 VE( X)  Finally, we verify the uniqueness of such an element c 3 VE( X). Let c 4  VE( X) be such that c 3 = c 4 , c 4  c 1 , and c 4  c 2 . Then since δ(c 3 , c 4 ) < +∞, it follows immediately from the above discussion that there exists an element  such that c 5  c 3 , and c 5  c 4 . Moreover, since c 1 = c 2 , and c 5 VE( X) c 3 = c 4 , by permuting {c 3 , c 4 } or {c 1 , c 2 } if necessary, we may assume without loss of generality that c 5 = c 3 , and c 3 = c 1 . On the other hand, the resulting 47 nontriviality of the specialization relations c 5  c 3  c 1 then contradicts the 1-dimensionality of the special fibers of the compactified semistable models “Z”  This completes the proof of assertion that appear in the definition of VE( X). (ix). def Next, we verify assertion (x). Write c 1 = c. Suppose that c 2  c 1 , c  2  c 1  \ {c 1 }. Then it follows that δ(c 2 , c  ) < for distinct elements c 2 , c  2 VE( X) 2 +∞. Thus, we conclude from Proposition 2.3, (ix), that there exists an element  such that c 3  c 2 , and c 3  c  . In particular, by permuting c 3 VE( X) 2 {c 2 , c  2 } if necessary, we may assume without loss of generality that c 3 = c 2 , and c 2 = c 1 . On the other hand, the resulting nontriviality of the specialization relations c 3  c 2  c 1 then contradicts the uniqueness portion of Proposition 2.3, (ix). This completes the proof of assertion (x). Next, we verify assertion (xi). Let Z X be a [connected] finite étale Ga-  Z X such that Z has split lois covering equipped with a factorization X stable reduction over K; Z a compactified semistable model with split reduc- tion of Z over O K that is stabilized by the natural action of Gal(Z/X). [Note that it follows immediately from Proposition 2.3, (iii), that such compactified semistable models form a directed inverse system that is cofinal in the directed  inverse system that appears in the definition of VE( X).] Write Z for the com- pactified stable model with split reduction of Z over O K ; Γ for the dual graph of Z s ; Γ for the dual graph of Z s . Observe that the natural action of s(H) on Γ factors through a finite quotient of s(H). Thus, it follows immediately from [CbTpIV], Corollary 1.15, (iii), that the natural action of s(H) on Γ has a fixed point c Γ. On the other hand, it follows immediately from the well-known theory of stable and semistable models [i.e., which may be reduced, by adding finitely many suitably positioned cusps, to the theory of pointed stable curves and contraction morphisms that arise from eliminating cusps, as exposed in [Knud]] that the inverse image of [the node or interior of an irreducible compo- nent in Z s corresponding to] c via the dominant morphism Z Z determines a tree inside Γ . Moreover, we recall that any action of a finite group on a tree has a fixed point [cf., e.g., [SemiAn], Lemma 1.8, (ii)]. Thus, we conclude that the natural action of s(H) on Γ has a fixed point. Since any inverse limit of nonempty finite sets is nonempty, we thus conclude that the natural action of  has a fixed point VE( X),  hence from Proposition 2.3, (x), s(H) on VE( X)   prim . that the natural action of s(H) on VE( X) prim has a fixed point VE( X) This completes the proof of assertion (xi). Next, we verify assertion (xii). Fix a prime number l Σ\{p}. Then observe that it follows from the stable reduction theorem [cf. [DM], [Knud]] that, after possibly replacing K by a suitable finite extension field of K, we may assume without loss of generality that Σ = {p, l}, and that X has stable reduction over K. Write X for the [unique, up to unique isomorphism] compactified stable model of X over O K . Next, observe that it follows immediately from Hurwitz’s formula, together with the well-known structure of geometric fundamental groups of hyperbolic curves over fields of characteristic zero [cf., e.g., [CmbGC], Remark 1.1.3], that, 48 after possibly replacing K by a suitable finite extension field of K, there exists a connected geometrically pro-p finite étale covering Y X of hyperbolic curves with split stable reduction over K that satisfies the condition that Y is of genus g Y 2. Write Y for the smooth compactification of Y over K; Y for the [unique, up to unique isomorphism] compactified stable model of Y over O K . Then observe that, if Y is smooth over O K , then it follows immediately from the non-injectivity of the natural surjective homomorphism ab Π ab Y Z/pZ  Π Y Z/pZ s [where we recall that, since Y is of genus g Y 2, the domain of this homo- morphism is of cardinality p 2g Y , while the codomain of this homomorphism is of cardinality p g Y ], together with Hurwitz’s formula [cf. also Zariski-Nagata purity; [ExtFam], Theorem A], that, after possibly replacing K by a suitable finite extension field of K, there exists a [connected] finite étale cyclic covering Y Y of hyperbolic curves over K that is of degree p and, moreover, satisfies the property that Y has bad reduction. In particular, by replacing Y × Y Y by Y , we may assume without loss of generality that Y has bad reduction. More- over, by replacing the connected geometrically pro-p finite étale covering Y X by its Galois closure, we may assume without loss of generality that Y X is a [connected] geometrically pro-p finite étale Galois covering [cf. Remark 2.1.3; Hurwitz’s formula]. Next, observe that it follows immediately from the theory of admissible cov- erings [cf., e.g., [Hur], §3], together with Hurwitz’s formula [and the well-known structure of geometric pro-l fundamental groups of hyperbolic curves over fields of characteristic p = l cf., e.g., [CmbGC], Remark 1.1.3], that, after possibly replacing K by a suitable finite extension field of K, there exists a [connected] geometrically pro-l finite étale Galois covering Z Y of hyperbolic curves with split stable reduction over K that satisfies the condition that every irreducible component of the special fiber of the [unique, up to unique isomorphism] sta- ble model of Z is a smooth curve of genus 2. Here, note that, by replacing Z Y by the composite of the Gal(Y /X)-conjugates of the admissible covering Z Y , we may assume without loss of generality that the composite covering def Z Y X is Galois. Thus, by taking X = Z, we obtain a [connected] geometrically pro-Σ finite étale Galois covering X X of hyperbolic curves satisfying the conditions in the statement of assertion (xii), as desired. This completes the proof of assertion (xii), hence of Proposition 2.3. Remark 2.3.1. We maintain the notation of Proposition 2.3. Then we observe that the statement of Proposition 2.3, (ii), becomes false if one omits the con- dition that the p-valuation v is residue-transcendental. Indeed, it suffices to construct an example of a discrete p-valuation on K(X) whose residue field is algebraic over the residue field of O K . Suppose that no finite extension field of the residue field of O K is separably closed [a condition that is satisfied if, for instance, the residue field of O K is finite]. Then one verifies immediately that 49  ur ) \ X(K) = ∅, and that for any x X( K  ur ) \ X(K), the point-theoretic X( K valuation associated to x on K(X) satisfies the desired properties. Remark 2.3.2. An alternative proof of Proposition 2.3, (iv), may be found in [Ray2], Proposition 5. The proof of Proposition 2.3, (iv), given in the present paper is of interest in that it involves techniques that are closer to the overall approach of the present paper.  an  tor of Proposition 2.3, (viii), Remark 2.3.3. The homeomorphism X VE( X) is essentially the same as the homeomorphism of [Lpg1], Proposition 1.1, but we give [essentially] self-contained statements and proofs here in the language of the present discussion for the sake of completeness. Remark 2.3.4. We maintain the notation of Proposition 2.3. Let X be a com- pactified semistable model of X over O K ; x X a smooth closed point. Write η for the generic point of the unique irreducible component of X s that contains x. Then one may construct a p-valuation v on K(X) associated to x by taking the ring of integers O v to consist of the elements K(X) that are integral with respect to the discrete valuation on K(X) associated to η and, moreover, map to an element in the residue field k(η) of X at η that is integral with respect to the discrete valuation on k(η) determined by x. Note that η determines a prime ideal of O v that contains p. In particular, v is nonprimitive. Proposition 2.4 (First properties of resolution of nonsingularities). Let Σ Primes be a nonempty subset; K a mixed characteristic complete discrete valuation field of residue characteristic p; X a hyperbolic curve over K. Then: (i) Let U X be an open subscheme [so U is a hyperbolic curve over K]. Suppose that X satisfies Σ-RNS. Then it holds that U satisfies Σ-RNS. (ii) Let f : Y X be a connected geometrically pro-Σ finite étale covering over K [so Y is a hyperbolic curve over a finite extension field of K]. Then it holds that X satisfies Σ-RNS if and only if Y satisfies Σ-RNS. (iii) Suppose that X satisfies the following condition: Let X be a compactified model of X over O K ; x X s a closed point. Then, after possibly replacing K by a suitable finite ex- tension field of K, there exist a connected geometrically pro-Σ finite étale Galois covering Y X of hyperbolic curves over K, a compactified semistable model Y of Y over O K , a morphism Y X of compactified models over O K that restricts to the finite étale Galois covering Y X, 50 an irreducible component D of Y s whose normalization is of genus 1, and whose image in X s is x X s . Then X satisfies Σ-RNS. (iv) Suppose that X satisfies Σ-RNS. Let X be a compactified model of X over O K . Then, after possibly replacing K by a suitable finite extension field of K, there exists a connected geometrically pro-Σ finite étale Galois covering Y X over K, together with a compactified stable model Y of Y over O K , such that the covering Y X extends to a morphism Y X . (v) Suppose that we are in the situation of Proposition 2.3, (v), and that X satisfies Σ-RNS. Write  st , VE( X)  st,tor , VE( X)  st,prim , VE( X)  st,pt-th VE( X)  VE( X)  tor , VE( X)  prim , VE( X)  pt-th for the modified versions of VE( X), obtained by requiring that the compactified semistable models “Z” that appear in the inverse limits used to define these sets be compactified stable models. [Here, we observe that, in light of (iv), the various toral compactified semistable models “Z relative to “Z” that appear in the construction of “VE(Z) tor in Definition 2.2, (vi), may be understood as being obtained as the result of contracting suitable irreducible components in the special fibers [cf. Remark 2.1.6] of suitable quotients of compactified stable models as in Proposition 2.3, (iv).] Then the natural maps  −→ VE( X)  st VE( X)  tor −→ VE( X)  st,tor VE( X)  prim −→ VE( X)  st,prim VE( X)  pt-th −→ VE( X)  st,pt-th VE( X) are bijective. (vi) Suppose that we are in the situation of Proposition 2.3, (v). Then X  it holds that satisfies Σ-RNS if and only if for each c VE( X), R c = lim O Z st ,z c , −→ st Z  denotes the valuation ring of the p-valuation associ- where R c K( X) ated to c [cf. Proposition 2.3, (v)]; the direct limit ranges over the set of compactified stable models with split reduction Z st of the domain curves of connected finite étale coverings Z X equipped with a factorization  Z X; z c denotes the center on Z st determined by R c ; the tran- X sition maps in the direct limit are induced by the corresponding scheme- theoretic morphisms of compactified stable models [which, in light of (iv), form a directed inverse system]. 51 (vii) Suppose that we are in the situation of Proposition 2.3, (v), and that X satisfies Σ-RNS. Let l Σ \ {p}; H G K a closed subgroup such that the intersection H I K of H with the inertia subgroup I K of G K (Σ) admits a surjection to [the profinite group] Z l ; s : H Π X a section (Σ) of the restriction to H of the natural surjection Π X  G K . Then there  prim that is fixed by the restriction, exists at most one element c VE( X) (Σ)  prim VE( X);  if, via s, to H of the natural action of Π X on VE( X) moreover, the restriction to H of the l-adic cyclotomic character of K  prim . In has open image, then there exists a unique such element c VE( X)  an particular, if X is proper, then there exists at most one element c an X [cf. Proposition 2.3, (vii)] that is fixed by the restriction, via s, to H of (Σ)  an ; if, the natural action of Π X on the topological pro-Berkovich space X moreover, the restriction to H of the l-adic cyclotomic character of  an . K has open image, then there exists a unique such element c an X Proof. Assertions (i), (ii) follow immediately from the various definitions in- volved. Next, we verify assertion (iii). Let v be a discrete residue-transcendental p-valuation on K(X). Then it follows immediately from Proposition 2.3, (ii), (iii), that, after possibly replacing K by a suitable finite extension field of K, there exists a compactified semistable model X of X over O K such that v arises from an irreducible component of X s . Let {x 1 , . . . , x N } X s be a fi- nite set of distinct closed points in the smooth locus of X s such that every irreducible component of X s whose normalization is of genus 0 contains three points {x 1 , . . . , x N }. Then it follows immediately from our assumption on X that, after possibly replacing K by a suitable finite extension field of K, for each positive integer i N , there exist a connected geometrically pro-Σ finite étale Galois covering Y i X over K, a morphism f i : Y i X of compactified semistable models over O K that restricts to the finite étale Galois covering Y i X, an irreducible component D i of (Y i ) s whose normalization is of genus 1, and whose image in X s is x i . Write f η : Y X for the connected geometrically pro-Σ finite étale Galois covering over K obtained by forming the composite of the finite étale Galois coverings {Y i X} 1≤i≤N over K. Then it follows immediately from Propo- sition 2.3, (iii), that, after possibly replacing K by a suitable finite extension field of K, there exists a compactified semistable model Y of Y over O K that dominates the respective normalizations of the semistable models {Y i } 1≤i≤N in the function field of Y . In particular, for each positive integer i N , there exists an irreducible component D i of (Y ) s whose normalization is of genus 1, and whose image in X s is x i . Next, let us observe that it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that, after possibly replacing K by a suit- able finite extension field of K, we may regard Y as the compactified stable 52 model associated to the hyperbolic curve Y obtained by removing from Y a collection of K-rational points of Y . In a similar vein, it follows immediately from the theory of pointed stable curves, as exposed in [Knud], that, after pos- sibly replacing K by a suitable finite extension field of K and replacing Y by a suitable Gal(Y /X)-stable dense open subscheme of Y , we may assume without loss of generality that Y is stabilized by the action of Gal(Y /X). Write f : Y X for the natural dominant morphism that restricts to the finite étale Galois cov- ering f η : Y X; κ : Y Y st for the natural dominant morphism to a compactified stable model Y st of Y over O K [cf. [ExtFam], Theorem A]. Thus, for each positive integer i N , f (D i ) = x i X s . In particular, since the covering f η : Y X is Galois, and Y is stabilized by the action of Gal(Y /X), it follows immediately from Zariski’s Main Theorem that, for each positive integer i N , the inverse image (f ) −1 (x i ) Y s is a closed subscheme that contains D i and is pure of dimension 1. Here, we recall that D i is an irreducible component of Y s whose normalization is of genus 1, hence necessarily maps birationally, via κ , to an irreducible component of Y st . In particular, we conclude that each connected component of (f ) −1 (x i ) Y s contains an irreducible component of Y s that maps birationally, via κ , to an irreducible component of Y st . Next, let D Y s be an irreducible component of Y s that maps to a closed point κ (D ) of Y s st via κ : Y Y st , but dominates an irreducible component def E = f (D ) of X s . Note that these assumptions imply that the normalization of D is of genus 0, and hence that E is an irreducible component of X s whose normalization is of genus 0. Thus, we conclude [cf. the condition imposed on the subset {x 1 , . . . , x N } X s ] that E contains three points {x 1 , . . . , x N }, i.e., [since (f ) −1 (x i ) is pure of dimension 1] that D contains at least 3 nodes [that map to three distinct “x i ”]. On the other hand, this [together with the birationality of κ ] implies that the closed point κ (D ) of Y s st intersects three distinct irreducible components of Y s st [i.e., the images of suitable irreducible components of (f ) −1 (x i ) Y s , for three distinct “i”], that is to say, in con- tradiction to the definition of the notion of a compactified stable model [cf. Definition 2.1, (iv)]. Thus, we conclude that there do not exist any such “D [i.e., that map to a closed point of Y s st , but dominate an irreducible component of X s ], and hence, by Zariski’s Main Theorem, that the morphism f : Y X factors as the composite of κ with a morphism f st : Y st X . In particular, it follows from the existence of the morphism f st : Y st X that Y s st contains an irreducible component whose corresponding valuation induces the given val- uation v on K(X), i.e., that X satisfies Σ-RNS. This completes the proof of assertion (iii). Next, we verify assertion (iv). In light of Proposition 2.3, (iii), after possibly replacing K by a suitable finite extension field of K, we may assume without loss 53 of generality that X is a compactified semistable model with split reduction of X over O K . Write {v 1 , . . . , v N } for the set of discrete valuations on K(X) that arise from the irreducible components of X s . Then since X satisfies Σ-RNS, for each positive integer i N , there exists a connected geometrically pro-Σ finite étale Galois covering Y i X such that v i coincides with the restriction of a discrete valuation on the function field of Y i that arises from an irreducible component of the special fiber of a compactified stable model of Y i . Write Y X for the composite covering of the connected geometrically pro-Σ finite étale Galois coverings {Y i X} 1≤i≤N . Then it follows immediately from Zariski’s Main Theorem [cf. also [ExtFam], Theorem A] that, after possibly replacing K by a suitable finite extension field of K, there exists a compactified stable model Y of Y over O K that dominates X . This completes the proof of assertion (iv). Assertion (v) and the necessity portion of assertion (vi) follow immediately from Proposition 2.4, (iv), together with Proposition 2.3, (iv), (vi). Next, we  be an element consider the sufficiency portion of assertion (vi). Let c VE( X)  that extends that corresponds [cf. Proposition 2.3, (v)] to a valuation v on K( X) a discrete residue-transcendental p-valuation on X. [Note that in this situation, v itself is necessarily residue-transcendental.] Then it suffices to show that there exists a “z c as in the statement of Proposition 2.4, (vi), that is a generic point of “Z s st ”. To this end, we observe that the nonexistence of such a “z c would imply that all of the “z c are closed points of “Z s st ”, hence have residue fields that are algebraic over the residue field of O K . On the other hand, this would imply that the residue field of R c = O v is algebraic over the residue field of O K , in contradiction to the residue-transcendentality of v. This completes the proof of assertion (vi). Finally, we verify assertion (vii). The portion of assertion (vii) concerning  prim as in the statement of Proposition the existence of an element c VE( X) 2.4, (vii), follows from Proposition 2.3, (xi). To verify the portion of assertion  prim , it suffices (vii) concerning the uniqueness of such an element c VE( X)  prim that satisfy the to show the equality of any two elements c 1 , c 2 VE( X) condition imposed on element “c” in the statement of Proposition 2.4, (vii).  st,prim . Then it follows from Write c  1 , c  2 for the images of c 1 , c 2 in VE( X) Proposition 2.4, (iv); [CbTpIV], Corollary 1.15, (iv) [applied to c  1 , c  2 ], that δ(c 1 , c 2 ) < +∞, hence from Proposition 2.3, (ix), that c 1 = c 2 , as desired. This completes the proof of assertion (vii), hence of Proposition 2.4. Corollary 2.5 (Constructions associated to geometric tempered fun- damental groups). Let Σ Primes be a subset of cardinality 2 such that p Σ; K , K mixed characteristic complete discrete valuation fields of residue characteristic p; X , X hyperbolic curves over K , K , respectively. Write Ω , Ω for the p-adic completions of K , K , respectively. For any hyperbolic curve Z over K , K , Ω , or Ω , write Π tp Z for the Σ-tempered fundamental group of Z, relative to a suitable choice of basepoint [cf. the subsection in Notations 54  X , X  X and Conventions entitled “Fundamental groups”]. Write X tp for the universal geometrically pro-Σ coverings corresponding to Π X , Π tp , re- X spectively. Suppose that X and X satisfy Σ-RNS. Then the following hold [cf. Remark 2.5.1 below]: (i) Let σ : Π tp Π tp be an isomorphism of topological groups. Then σ X X induces homeomorphisms  ),  ) VE( X VE( X  ) tor  ) tor , VE( X VE( X  ) prim VE( X  ) prim , VE( X that are compatible with the respective natural actions of Π tp , Π tp . If, X X moreover, X and X are proper, then σ induces a homeomorphism  ) an  ) an ( X ( X that is compatible with the respective natural actions of Π tp , Π tp . X X def def (ii) Suppose that K = K = K , K = K = K , hence that Ω = Ω = Ω . Let x X (Ω), x X (Ω). Write X x (respectively, X x ) for the  : Π tp hyperbolic curve X Ω \{x } (respectively, X Ω \{x }) over Ω. Let σ X x tp Π be an isomorphism of topological groups that fits into a commutative X x diagram Π tp −−−−→ Π tp X X σ  x x   Π tp −−−−→ Π tp , X X σ where the vertical arrows are the natural surjections [determined up to composition with an inner automorphism] induced by the natural open immersions X x → X Ω , X x → X Ω of hyperbolic curves; the lower hori- zontal arrow σ is the isomorphism of topological groups [determined up to composition with an inner automorphism] induced by a(n) [uniquely de- termined cf., e.g., [DM], Lemma 1.14] isomorphism σ X : X X of schemes over K. Then x = σ X (x ). Proof. First, we verify assertion (i). We begin by recalling that [SemiAn], Corol- lary 3.11, may be generalized/applied to hyperbolic curves over an arbitrary mixed characteristic complete discrete valuation field [cf. [AbsTopII], Remark 2.11.1, (i)]. Thus, by applying this generalized version of [SemiAn], Corollary  ) st .  ) st VE( X 3.11, we conclude that σ induces a homeomorphism VE( X On the other hand, it follows from our assumption that X and X satisfy   st and Σ-RNS that we may apply the homeomorphisms “VE( X) VE( X) 55  tor  st,tor of Proposition 2.4, (v). In particular, we conclude “VE( X) VE( X) that σ induces a homeomorphism  ) tor  ) tor VE( X VE( X that is manifestly compatible with the respective natural actions of Π tp , Π tp , X X as well as a homeomorphism  )  ) VE( X VE( X , Π tp that is manifestly compatible with the respective natural actions of Π tp X X and preserves specialization/generization relations, hence induces a homeomor- phism  ) prim  ) prim VE( X VE( X that is compatible with the respective natural actions of Π tp , Π tp . Finally, if, X X moreover, X and X are proper, then we may apply the homeomorphism “θ X  of Proposition 2.3, (viii), to conclude that σ induces a homeomorphism  ) an  ) an ( X ( X that is compatible with the respective natural actions of Π tp , Π tp . This com- X X pletes the proof of assertion (i). Next, we verify assertion (ii). We begin by observing that it follows from the generalized version of [SemiAn], Corollary 3.11, discussed above, together with Corollary 2.5, (i), that σ  induces a bijection  )  ) VE( X VE( X that maps the point-theoretic orbit-center-chain associated to x to the point- theoretic orbit-center-chain associated to x . Since σ arises from σ X , we thus  pt-th /Gal( X/X  conclude from the bijection “X(Ω) VE( X) K )” of Proposition 2.3, (v), that σ X (x ) = x . This completes the proof of assertion (ii), hence of Corollary 2.5.  ) an  ) an of Corollary 2.5, (i), Remark 2.5.1. The homeomorphism “( X ( X is essentially similar to the homeomorphisms of [Lpg1], Theorem 3.10, but is formulated and proven according to the approach of the present paper. On the other hand, Corollary 2.5, (ii), may be regarded, when taken together with Theorem 2.17 below, as a generalization of [Tsjm], Theorem 2.2; its proof may be regarded as a more sophisticated version of the argument applied in the proof of [Tsjm], Theorem 2.2. Proposition 2.6 (Existence of new ordinary parts of certain cover- ings after Raynaud-Tamagawa). Let l be a prime number = p; K a mixed characteristic complete discrete valuation field of residue characteristic p; X a 56 proper hyperbolic curve over K. Suppose that X has split stable reduction over K. Write X st for the [unique, up to unique isomorphism] stable model of X over O K . Suppose, moreover, that every irreducible component of the special fiber of X st is a smooth curve of genus 2. Write e X (respectively, v X ) for the cardinality of the set of nodes (respectively, the set of irreducible components) of the stable curve X s st . Then: (i) For each sufficiently large positive integer m, if we replace K by a finite unramified extension field of K, then there exists a finite étale cyclic cov- ering Y st −→ X st over O K of degree l m satisfying the following conditions: (a) Write (Y st →) Z st X st for the finite étale cyclic subcovering over O K of degree l m−1 ; Y , Z for the generic fibers of Y st , Z st , respec- tively. Then Y and Z have split stable reduction over K. Moreover, Y st , Z st are the stable models of Y , Z, respectively. (b) The finite étale covering Y s st X s st determined by the finite étale cyclic covering Y st X st induces a bijection between the respective sets of irreducible components. (c) Write A for the abelian variety over K obtained by forming the coker- nel of the natural morphism J(Z) J(Y ) induced by the finite étale cyclic covering Y Z [of degree l]. Then there exists an abelian va- riety B over K with good ordinary reduction such that T p A fits into exact sequences of G K -modules [cf. the theory of [FC], especially, [FC], Chapter III, Corollary 7.3] 0 −→ T gd −→ T p A −→ T cb −→ 0 0 −→ T tor −→ T gd −→ T p B −→ 0 0 −→ Hom(T p B s , Z p (1)) −→ T p B −→ T p B s −→ 0, where “(1)” denotes the Tate twist; the natural action of G K on the “combinatorial quotient” T cb [i.e., the inverse limit of the quotients “Y /nY of [FC], Chapter III, Corollary 7.3, as n ranges over the positive integral powers of p] of T p A is trivial; T tor is isomorphic as a G K -module to the direct sum of a collection of copies of Z p (1); B denotes the abelian scheme over O K whose generic fiber is equal to B. (ii) Fix a finite étale cyclic covering Y st X st as in (i). Write T cb,Y , T cb,Z for the “combinatorial quotients” [i.e., the inverse limit of the quotients “Y /nY of [FC], Chapter III, Corollary 7.3, as n ranges over the positive integral powers of p] of T p J(Y ), T p J(Z), respectively; h Y , h Z for the 57 respective loop-ranks of the dual graphs associated to the stable curves Y s st , Z s st . Then it holds that h Y = 1 + l m e X v X , h Z = 1 + l m−1 e X v X . Moreover, rank Z p T cb,Y = h Y ; rank Z p T cb,Z = h Z ; rank Z p T tor = rank Z p T cb = h Y h Z = (l m l m−1 )e X . Proof. First, we verify assertion (i). Write {C i } 1≤i≤v X for the set of irreducible components of X s st . Let m be a positive integer such that, for each positive integer i v X , it holds that l m > l 2g Ci l 2g Ci −1 (p 1)g C i , l 2g Ci 1 where g (−) denotes the genus of (−). Then, in light of [Tama1], Lemma 1.9 [i.e., a generalization of [Ray1], Théorème 4.3.1], by replacing K by a finite unramified extension field of K, one may construct finite étale cyclic coverings {D i C i } 1≤i≤v X of degree l m [of proper hyperbolic curves over the residue field of K] satisfying the following conditions: For each positive integer i v X , write (D i →) E i C i for the finite étale cyclic subcovering of degree l m−1 [of proper hyperbolic curves over the residue field of K]. Then the abelian variety obtained by forming the cokernel of the natural morphism J(E i ) J(D i ) induced by the finite étale cyclic covering D i E i of degree l is ordinary. The cardinality of the set of closed points of D i that lie over the closed points of C i determined by the nodes of X s st is equal to l m . Next, one verifies immediately that there exists a finite étale cyclic covering D X s st of degree l m obtained by gluing together the finite étale cyclic cov- erings {D i C i } 1≤i≤v X . Write Y st X st for the finite étale cyclic covering obtained by deforming the finite étale cyclic covering D X s st . Then it follows immediately from the various definitions involved that conditions (a), (b) hold. Moreover, in light of the theory of Raynaud extensions [cf. [FC], Chapter II, §1; [FC], Chapter III, Corollary 7.3], together with Remark 2.6.1, (i), (ii), below [cf. also [BLR], §9.2, Example 8], one concludes that condition (c) holds. This completes the proof of assertion (i). Next, we verify assertion (ii). Write e Y , e Z for the respective cardinalities of the sets of nodes of the stable curves Y s st , Z s st ; v Y , v Z for the respective cardinalities of the sets of irreducible components of the stable curves Y s st , Z s st . Then it follows immediately from conditions (a), (b), that e Y = l m e X , e Z = l m−1 e X , 58 v Y = v Z = v X . Thus, we conclude from the well-known computation of the first homology group of a finite graph that h Y = 1 + e Y v Y = 1 + l m e X v X , h Z = 1 + e Z v Z = 1 + l m−1 e X v X . Therefore, to complete the proof of assertion (ii), it suffices to prove that rank Z p T cb,Y = h Y , rank Z p T cb,Z = h Z , and rank Z p T tor = rank Z p T cb = h Y −h Z . Recall that the loop-ranks h Y , h Z coincide with the toric ranks of the Jacobians of the stable curves Y s st , Z s st , respectively [cf., e.g, [BLR], §9.2, Example 8]. On the other hand, in light of the theory of duality for torsion subgroups of abelian varieties, it holds that these toric ranks coincide with the ranks of the respective corresponding combinatorial quotients [cf. [FC], Chapter III, Corollary 7.4]. In particular, it follows immediately [cf. Remark 2.6.1, (ii), below; [FC], Chap- ter III, Corollary 7.4] that rank Z p T cb,Y = h Y , rank Z p T cb,Z = h Z , hence that rank Z p T tor = rank Z p T cb = rank Z p T cb,Y rank Z p T cb,Z = h Y h Z . This completes the proof of assertion (ii), hence of Proposition 2.6. Remark 2.6.1. We maintain the notation of Proposition 2.6. (i) Write A for the identity component of the Néron model of A over O K [cf. [BLR], §1.3, Corollary 2]. Then the universal property of the Néron model implies the existence of a surjective homomorphism f : Pic 0 Y st /O K  A that extends the natural quotient homomorphism J(Y )  A [cf. [BLR], §1.2, Definition 1; [BLR], §9.4, Theorem 1]. Thus, since Pic 0 Y st /O K is a semi-abelian scheme over O K [cf. [BLR], §9.4, Theorem 1], it follows immediately from the existence of the surjective homomorphism f that A is also a semi-abelian scheme over O K . (ii) Recall that the composite homomorphism J(Z) J(Y ) J(Z) of the norm map J(Y ) J(Z) with the natural homomorphism J(Z) J(Y ) coincides with the morphism given by multiplication by l. In particular, the abelian variety J(Y ) is isogenous over K to the product abelian variety J(Z) × K A. Thus, we conclude from [BLR], §7.3, Proposition 6, that the semi-abelian schemes Pic 0 Y st /O K and Pic 0 Z st /O K × O K A over O K [cf. [BLR], §9.4, Theorem 1] are isogenous over O K . Definition 2.7. In the notation of Remark 2.6.1, let Z be a semistable model of Z over O K that has split reduction. Note that Z st satisfies this property, and that, by pulling-back the finite étale cyclic covering Y st Z st via the unique morphism Z Z st that extends the identity morphism Z Z [cf. [ExtFam], Theorem A], we obtain a finite étale cyclic covering Y −→ Z over O K of degree l that extends the finite étale cyclic covering Y Z over K. Suppose that 59 X s st is a singular curve, and that m is sufficiently large that h Y h Z 1 [cf. Proposition 2.6, (ii)]. (i) Let y η Y (K). Write z η Z(K) for the image of y η via the natural map Y (K) Z(K). Then y η and z η determine embeddings Y → J(Y ), Z → J(Z) that allow one to regard J(Y ), J(Z) as the respective Albanese varieties of Y , Z [cf. [AbsTopI], Appendix, Definition A.1, (ii); [Milne], Proposition 6.1]. In particular, we obtain a commutative diagram Δ Y −−−−→ T p J(Y ) −−−−→ T cb,Y    Δ Z −−−−→ T p J(Z) −−−−→ T cb,Z , where the left-hand vertical arrow denotes the open injection induced by the finite étale cyclic covering Y Z; the left-hand horizontal arrows denote the natural surjections determined by the Albanese embeddings Y → J(Y ), Z → J(Z) [cf. [AbsTopI], Appendix, Proposition A.6, (iv)]; the right-hand horizontal arrows denote the natural surjections; the mid- dle and right-hand vertical arrows are surjections [cf. the fact that the finite étale cyclic covering Y Z is of degree l = p]; the first square of the diagram commutes in light of the functoriality of the étale funda- mental group; the second square of the diagram commutes in light of the functoriality of Raynaud extensions. (ii) Fix a quotient T cb,Z  Z p [cf. our assumption that h Z 1; Proposition 2.6, (ii)]. For each nonneg- ative integer n, write Z n −→ Z for the finite étale cyclic [“combinatorial”] covering of degree p n over O K induced by the natural quotient Z  T p J(Z) ) T cb,Z  Z p  Z/p n Z; Y n −→ Y for the finite étale cyclic [“combinatorial”] covering of degree p n over O K induced by the natural quotient Y  T p J(Y ) ) T cb,Y  T cb,Z  Z p  Z/p n Z. Thus, the commutative diagram in (i) induces a cartesian commutative diagram Y n −−−−→ Y   Z n −−−−→ Z, 60 where the vertical arrows are finite étale cyclic coverings of degree l over O K ; the horizontal arrows are finite étale cyclic [“combinatorial”] cover- ings of degree p n over O K . Moreover: (a) Write Y n , Z n for the generic fibers of Y n , Z n , respectively. Then the finite étale covering (Y n ) s (Z n ) s determined by the finite étale cyclic covering Y n Z n induces a bijection between the sets of irreducible components that arise from the respective stable models of Y n and Z n [cf. Proposition 2.6, (i), (b)]. (b) Write A n for the abelian variety over K obtained by forming the cokernel of the natural morphism J(Z n ) J(Y n ) induced by the finite étale cyclic covering Y n Z n of degree l [of proper hyperbolic curves over K]. Then there exists an abelian variety B n over K with good ordinary reduction such that T p A n fits into exact sequences of G K -modules [cf. the theory of [FC], especially, [FC], Chapter III, Corollary 7.3; the proof of Proposition 2.6, (i), (c)] 0 −→ T gd,n −→ T p A n −→ T cb,n −→ 0 0 −→ T tor,n −→ T gd,n −→ T p B n −→ 0 0 −→ Hom(T p (B n ) s , Z p (1)) −→ T p B n −→ T p (B n ) s −→ 0, where “(1)” denotes the Tate twist; the natural action of G K on the “combinatorial quotient” T cb,n of T p A n is trivial; T tor,n is isomorphic as a G K -module to the direct sum of a collection of copies of Z p (1); B n denotes the abelian scheme over O K whose generic fiber is equal to B n . (iii) Write A n for the identity component of the Néron model of A n over O K [cf. [BLR], §1.3, Corollary 2]. Then the universal property of the Néron model implies the existence of a surjective homomorphism def f n : J n = Pic 0 Y n /O K  A n that extends the natural quotient homomorphism J(Y n )  A n [cf. [BLR], §1.2, Definition 1; [BLR], §9.4, Theorem 1]. Thus, since J n = Pic 0 Y n /O K is a semi-abelian scheme over O K [cf. [BLR], §9.4, Theorem 1], it follows immediately from the existence of the surjective homomorphism f n that A n is also a semi-abelian scheme over O K [cf. Remark 2.6.1, (i)]. (iv) For each nonnegative integer n, write h Y n , 61 h Z n for the loop-ranks of the dual graphs associated to the semistable curves (Y n ) s , (Z n ) s , respectively; g Y n for the [arithmetic] genus of the semistable model Y n over O K . Then a similar argument to the argument applied in Remark 2.6.1, (ii), implies that the semi-abelian schemes Pic 0 Y n /O K and Pic 0 Z n /O K × O K A n over O K are isogenous over O K . In particular, we obtain equalities rank Z p T tor,n = rank Z p T cb,n = h Y n h Z n [cf. the proof of Proposition 2.6, (ii)]. Proposition 2.8 (Explicit computations of toric rank and genus). We maintain the notation of Definition 2.7. Then the following hold: (i) It holds that h Y n = 1 + p n l m e X p n v X , h Z n = 1 + p n l m−1 e X p n v X , hence, in particular, that rank Z p T tor,n = rank Z p T cb,n = h Y n h Z n = p n (l m l m−1 )e X [cf. the final display of Definition 2.7, (iv)]. (ii) It holds that g Y n = p n (g Y 0 1) + 1. Proof. First, recall from the well-known theory of stable and semistable models that h Y n , h Z n , rank Z p T tor,n , rank Z p T cb,n , and g Y n are independent of the choice of the semistable model Z of Z over O K . [Indeed, by passing to a suitable finite unramified extension of K and adding finitely many suitably positioned cusps, one may, in effect, reduce this “well-known theory of stable and semistable models” to the theory of pointed stable curves and contraction morphisms that arise from eliminating cusps, as exposed in [Knud].] In particular, we may assume without loss of generality that Z is the stable model of Z over O K . Assertion (i) then follows immediately from a similar argument to the argument applied in the proof of Proposition 2.6, (ii). Assertion (ii) follows immediately from Hurwitz’s formula. This completes the proof of Proposition 2.8. Definition 2.9. We maintain the notation of Definition 2.7. Then we shall write T  cnn,n 62 for the connected p-divisible group over O K that arises as the connected part of the p-divisible group [cf. the discussion preceding [Tate], §2.2, Proposition 2] associated to the Raynaud extension [cf. [FC], Chapter II, §1] of A n ; T  tor,n for the connected p-divisible group over O K associated to the torus that appears in the Raynaud extension of A n ; T  cnn,n , T  tor,n for the respective generic fibers of T  cnn,n , T  tor,n ; def T cnn,n = T p ( T  cnn,n ), def T tor,n = T p ( T  tor,n ) for the respective p-adic Tate modules of T  cnn,n , T  tor,n [cf. the subsection in Notations and Conventions entitled “Schemes”]. Note that T cnn,n and T tor,n may be regarded as Z p -submodules of T p A n in a natural way. Moreover, we shall write def T ét,n = T p A n /T cnn,n . Note that G K acts naturally on the Z p -modules T cnn,n , T tor,n , T ét,n , and T cb,n [cf. Definition 2.7, (ii), (b)]. We shall write T cnn,n  T qtr,n for the maximal torsion-free G K -stable quotient Z p -module among the torsion- free G K -stable quotient Z p -modules T cnn,n  T such that some open subgroup of G K acts on T via the p-adic cyclotomic character; T qcb,n T ét,n for the maximal G K -stable Z p -submodule among the G K -stable Z p -submodules T T ét,n such that some open subgroup of G K acts trivially on T . Finally, we observe that [one verifies immediately that] we obtain natural exact sequences of G K -modules [cf. Definition 2.7, (ii), (b)] 0 −→ T tor,n −→ T cnn,n −→ Hom(T p (B n ) s , Z p (1)) −→ 0 0 −→ T p (B n ) s −→ T ét,n −→ T cb,n −→ 0 0 −→ T cnn,n −→ T p A n −→ T ét,n −→ 0. Lemma 2.10. We maintain the notation of Definition 2.9. Then: (i) The natural action of G K on T p A n induces the trivial action of I K on T ét,n . 63 (ii) There exist natural compatible G K -equivariant isomorphisms T ét,n Hom(T cnn,n , Z p (1)), T cb,n Hom(T tor,n , Z p (1)), T qcb,n Hom(T qtr,n , Z p (1)). (iii) Suppose that K is a p-adic local field. Then the set of eigenvalues of the Z p -linear automorphism of T p (B n ) s induced by the Frobenius element of G K /I K [cf. (i); the second exact sequence of Definition 2.9] does not contain any roots of unity. Proof. First, we verify assertion (i). Recall that the quotient of a p-divisible group by its connected part is étale [cf., e.g., the discussion preceding [Tate], §2.2, Proposition 2]. Thus, we conclude [cf. the triviality of the action of G K on T cb,n observed in Definition 2.7, (ii), (b); the second exact sequence of the final display of Definition 2.9] from [the second sentence of] [FC], Chapter III, Corollary 7.3, that the natural action of I K on T ét,n is trivial, as desired. This completes the proof of assertion (i). Assertion (ii) follows immediately from the theory of duality for torsion subgroups of abelian varieties [cf. [FC], Chapter III, Corollary 7.4], together with the first and second exact sequences of the final display of Definition 2.9. Assertion (iii) follows immediately from the finiteness of the set of rational points of (B n ) s over any finite extension field of the [finite!] residue field of K. This completes the proof of Lemma 2.10. Proposition 2.11 (Toral quotient of the connected part). In the notation of Definition 2.9, write χ n : T cnn,n  T qtr,n for the natural surjection of G K -modules. Then the following hold: (i) Suppose that the residue field of K is separably closed. Then T cnn,n = T qtr,n = {0}, T qcb,n = T ét,n = {0}. (ii) Suppose that K is a p-adic local field. Then the restriction of χ n to T tor,n T cnn,n induces an injection T tor,n → T qtr,n with finite cokernel. Proof. First, we verify assertion (i). Note that since the residue field of K is separably closed, G K = I K . Thus, it follows immediately from Lemma 2.10, (i), together with the various definitions involved, that T qcb,n = T ét,n . On the other hand, it follows immediately from Proposition 2.8, (i) [cf. also Proposition 2.6; Definition 2.7; the second exact sequence of the final display of Definition 2.9], that T ét,n = {0}. Thus, we conclude from Lemma 2.10, (ii), that T cnn,n = T qtr,n = {0}. This completes the proof of assertion (i). def Next, we verify assertion (ii). Write N = rank Z p T qtr,n . Note that, in light of the maximality of T qtr,n , it suffices to verify that there exists a unique torsion-free G K -stable quotient Z p -module T cnn,n  T 64 whose restriction to T tor,n T cnn,n induces an injection T tor,n → T with finite cokernel, and that rank Z p T N . Let Q p be an algebraic closure of Q p equipped with the trivial action of G K . Then observe that it follows from a routine argument involving Galois descent from Q p to Q p that it suffices to verify that there exists a unique G K -stable quotient Q p -vector space T cnn,n Z p Q p −→ V whose restriction to T tor,n Z p Q p T cnn,n Z p Q p induces an isomorphism T tor,n Z p Q p V , and that dim Q p V N . Write k for the finite residue field of the p-adic local field K [so G K /I K may be identified with the absolute Galois group G k of k]. Then, by applying Lemma 2.10, (i), (ii), we conclude that it suffices to verify that there exists a unique G k -stable Q p -subspace V T ét,n Z p Q p whose composite with the natural surjection T ét,n Z p Q p  T cb,n Z p Q p induces an isomorphism V T cb,n Z p Q p , and that dim Q p V N . On the other hand, in light of the eigenspace decomposition associated to the natural action of the Frobenius element G k , the existence and uniqueness of such a subspace, together with the inequality dim Q p V N (= rank Z p T qcb,n ) [cf. Lemma 2.10, (ii)], follows immediately from Lemma 2.10, (iii) [cf. also the triviality of the action of G K on T cb,n observed in Definition 2.7, (ii), (b)]. This completes the proof of assertion (ii), hence of Proposition 2.11. Proposition 2.12 (Construction of a certain morphism of formal schemes to the quasi-toral quotient). In the notation of Proposition 2.11, let y n Y n (O K ) be an O K -rational point that maps the closed point of Spec O K to a smooth point (y n ) s of the semistable curve (Y n ) s . Write y n,η Y n (K) for the K-valued point of Y n determined by y n Y n (O K ); C (Y n ) s for the unique irreducible component that contains (y n ) s ; F Y n for the closed subset obtained by forming the union of the irreducible components = C of (Y n ) s ; U y n Y n for the open subscheme obtained by forming the complement of F Y n ; h n,η : Y n −→ J(Y n ) for the Albanese map that maps y n,η to the origin [cf. [AbsTopI], Appendix, Definition A.1, (ii); [Milne], Proposition 6.1]. Recall that J n is a semi-abelian scheme over O K [cf. Definition 2.7, (iii)] whose generic fiber is J(Y n ). In particular, J n is isomorphic to the identity component of the Néron model over O K of J(Y n ) [cf. [BLR], §7.4, Proposition 3]. Thus, since U y n is a connected smooth scheme over O K whose generic fiber is Y n , the universal property of the Néron model implies the existence of a unique morphism h n : U y n −→ J n that extends h n,η . Next, write J  n , A  n for the formal completions at the origin of  Y ,y , O  U ,y for the completions the semi-abelian schemes J n , A n over O K ; O n n yn n 65 at y n of O Y n ,(y n ) s , O U yn ,(y n ) s . Then the natural composite map  Y ,y = Spf O  U ,y −→ J  n −→ A  n Spf O n n yn n induced by h n and the surjective homomorphism f n : J n  A n [cf. Definition 2.7, (iii)] determines a morphism of formal O K -schemes  Y ,y −→ T  cnn,n , Spf O n n where we regard the connected p-divisible group T  cnn,n as a formal group over O K [cf. [Tate], §2.2, Proposition 1]. In particular, by forming the composite with the morphism of formal O K -schemes induced by χ n [cf. [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1], we obtain a morphism of formal O K -schemes  Y ,y −→ T  qtr,n , Spf O n n where T  qtr,n denotes the formal group over O K determined by the connected p- divisible group associated to the G K -module T qtr,n [cf. [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1]. Proof. Proposition 2.12 follows immediately from the various references quoted in the statement of Proposition 2.12. Proposition 2.13 (Coverings associated to characters). We maintain the notation of Proposition 2.12. Then the following hold:  m for the formal completion at the origin of the multiplicative (i) Write G  m ) for the Z p -module of group scheme G m over O K ; Hom O K ( T  qtr,n , G   homomorphisms over O K from T qtr,n to G m ; Hom G K (T qtr,n , Z p (1)) for the Z p -module of G K -equivariant homomorphisms of Z p -modules T qtr,n Z p (1). Then the natural homomorphism  m ) −→ Hom G (T qtr,n , Z p (1)) Hom O K ( T  qtr,n , G K is bijective.  m ). Consider the com- (ii) Let a be a positive integer; f Hom O K ( T  qtr,n , G posite  m  Y ,y −→ G (Spf O K [[t]] →) Spf O n n [where t is an indeterminate, and we regard O K [[t]] as being equipped with the t-adic topology] of the morphism in the final display of Proposition 2.12 with f . By a slight abuse of notation, we shall also write f O K [[t]] ×  m via the homomorphism for the image of the canonical coordinate U of G of rings induced by the above composite morphism. Then the covering of Spf O K [[t]] obtained by extracting a p a -th root of f is dominated by the covering of Spf O K [[t]] obtained by restricting the covering determined by multiplication by p a on A  n . 66 Proof. Assertion (i) follows immediately from [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1. Assertion (ii) follows immediately from the various definitions involved. This completes the proof of Proposition 2.13. Lemma 2.14. Let X be a smooth proper curve of genus g X over a field K; x X(K) a K-valued point of X; d x a nonnegative integer such that d x 2g X 1. Then the natural composite map H 0 (X, Ω X ) → Ω X,x  Ω X,x /m d x x Ω X,x is injective. Proof. First, observe that since d x 2g X 1 > 2g X 2 [which implies that the degree of the line bundle Ω X (−d x ·x) is negative], it follows that H 0 (X, Ω X (−d x · x)) = 0. Thus, the desired injectivity follows immediately by applying the [left exact] functor H 0 (X, −) to the short exact sequence 0 −→ Ω X (−d x ) −→ Ω X −→ Ω X,x /m d x x Ω X,x −→ 0. This completes the proof of Lemma 2.14. Lemma 2.15. Let a, b be positive integers; K a p-adic local field of degree def d K = [K : Q p ] over Q p ; W K[T ]/(T b+1 ) a Q p -vector subspace of dimension a. For each nonzero element  h i T i K[T ]/(T b+1 ), h = 0≤i≤b def write ord(h) (≤ b) for the smallest integer i such that h i = 0. Set ord(0) = +∞. Suppose that def W \{0} F = {h K[T ]/(T b+1 ) | ord(h) = p j −1 for some nonnegative integer j}. Then it holds that a d K (log p (b + 1) + 1)    d K (b + 1) = dim Q p K[T ]/(T b+1 ) . Proof. For each nonnegative integer j, write def F j = W {h K[T ]/(T b+1 ) | ord(h) p j 1} [so F j is a Q p -vector space, and F j+1 F j ]. Then it follows immediately from our assumption that W \ {0} F that for each nonnegative integer j, dim Q p (F j /F j+1 ) dim Q p (K) = d K . 67 On the other hand, since F j = {0} for any nonnegative integer j such that p j > p j 1 b + 1, we thus conclude that a = dim Q p (W ) = +∞  dim Q p (F j /F j+1 ) d K (log p (b + 1) + 1), j=0 as desired. This completes the proof of Lemma 2.15. Theorem 2.16 (Existence of suitable coverings). In the notation of Propo- sition 2.13, suppose further that K is a p-adic local field. Then there exists a def def real number C g Y ,d K that depends only on g Y = g Y 0 and d K = [K : Q p ] such that for any positive integer n C g Y ,d K , after possibly replacing K by a finite extension field of K, there exist a [connected] finite étale Galois covering W n Y n of proper hyperbolic curves over K of degree a power of p, a semistable model W n of W n over O K , a morphism ψ : W n Y n of semistable models over O K that restricts to the finite étale Galois covering W n Y n , an irreducible component D of (W n ) s whose normalization is of genus 1 such that ψ(D) = (y n ) s (Y n ) s . Proof. Fix a positive integer n. First, we consider the natural homomorphisms of Z p -modules   Hom G K (T qtr,n , Z p (1))  m ) → H 0 ( T  qtr,n , Ω  Hom O K ( T  qtr,n , G inv T qtr,n ) 0 ( T  cnn,n , Ω T  cnn,n ) → H inv H 0 (A n , Ω A n ) → H 0 (U y n , Ω Y n )  Y ,y → Ω Y n ,y n O Y n ,yn O n n 2g Yn Ω Y n ,y n,η /m y n,η Ω Y n ,y n,η   2g Yn K[t]/(t ) dt , where the first arrow denotes the natural bijective homomorphism of Proposition 2.13, (i); 68 0 “H inv (−)” denotes the O K -submodule of “H 0 (−)” that consists of the invariant differentials on the p-divisible group in the first argument of “H 0 (−)”; the second arrow denotes the injection obtained by pulling back the in- def  variant differential d log(U ) = dU U on G m ; the third arrow denotes the injection induced by χ n [cf. Proposition 2.11; [Tate], §2.2, Proposition 1; [Tate], §4.2, Corollary 1]; the fourth arrow denotes the natural isomorphism; the fifth arrow denotes the homomorphism of O K -modules obtained by pulling back the differentials via the composite map U y n → J n  A n that maps y n to the origin [cf. Definition 2.7, (ii), (b); Definition 2.7, (iii); Proposition 2.12]; the sixth arrow denotes the natural injection; the seventh arrow denotes the natural restriction morphism; m y n,η denotes the maximal ideal of O Y n ,y n,η ; the final arrow denotes the natural isomorphism determined by choosing a “local coordinate” t, i.e., an element of the maximal ideal m Y n ,y n of O Y n ,y n such that t and m K generate m Y n ,y n . Write  m ) −→ O K [[t]] × Ψ : Hom O K ( T  qtr,n , G for the assignment discussed in Proposition 2.13, (ii) [i.e., relative to the local coordinate t chosen above];   Yn  m ) → Ω Y ,y /m 2g K[t]/(t 2g Yn ) dt Ξ : Hom O K ( T  qtr,n , G y n,η Ω Y n ,y n,η n n,η for the injective [by Lemma 2.14] composite of the second to the seventh arrows in the first display of the present proof. Thus, Ξ may be understood as the result of composing Ψ with the operation of taking the logarithmic derivative with respect to t and then truncating the terms of degree 2g Y n . Observe that so far we have not applied the assumption that K is a p-adic local field. Now we proceed to apply this assumption. Recall from Proposition 2.8, (i), (ii); Proposition 2.11, (ii) [cf. also the initial portions of Proposition 2.6 and Definition 2.7], that rank Z p T qtr,n = rank Z p T tor,n = p n (l m l m−1 )e X ; 2g Y n = 2(p n (g Y 0 1) + 1), where e X 1. Thus, one verifies immediately that there exists a real number C g Y ,d K that depends only on g Y = g Y 0 and d K = [K : Q p ] such that for any positive integer n C g Y ,d K , it holds that p n (l m l m−1 )e X > d K (log p (2(p n (g Y 0 1) + 1)) + 1). 69 In particular, we conclude from Lemma 2.15 that there exists a homomorphism  m ) such that Ξ(f ) = 0, and ord(Ξ(f )) + 1 is not a [non- f Hom O K ( T  qtr,n , G negative integral] power of p. Fix such a homomorphism f . Then note that it follows from our choice of f that we may write  a i t i O K [[t]] × Ψ(f ) = 1 + i≥1 [cf. Proposition 2.13, (ii)], where, if we write i 0 for the smallest positive integer i such that a i = 0, then i 0 is not a [nonnegative integral] power of p. In the following, we shall apply Proposition 1.6, where we take “g(t)” to be Ψ(f ) and apply the isomorphism of topological O K -algebras  Y ,y O K [[t]] O n n def determined by t, to complete the proof of Theorem 2.16. Write N = μ + 1, where μ is the “μ” that results from applying Proposition 1.6, (i); ψ η : W n −→ Y n for the [connected] finite étale Galois covering over K obtained by pulling-back the morphism induced by multiplication by p N on A n via the composite mor- phism Y n → U y n J n  A n [cf. Proposition 2.12]. Next, let us observe that it follows immediately from the various definitions of the morphisms involved that the composite morphism φ  c λ h τ φ c | U : U c Spec O K [[t]] 1 2 c 2 2 [cf. the two commutative diagrams of Proposition 1.6, (i)], together with the   isomorphism O K [[t]] O Y n ,y n , allow one to regard the p-adic completion R Y of Γ(O U c 2 , U c 2 ) at the generic point of (U c 2 ) s as the p-adic completion of the ring of integers R Y of a certain discrete residue-transcendental p-valuation on the function field of Y n . Write R W/Y for the normalization of R Y in the function field of W n . Thus, since R Y is [a localization of a ring of finite type over the complete discrete valuation ring O K , hence] excellent, it follows that R W/Y is finite over R Y . Next, let us observe that it follows from the relations ι λ g φ  c λ h = (p μ ) ξ g , 1 ξ g τ c 2 | U c 2 ) = ι π p | U πp ) θ g , ι π p | U πp ) θ g f Y = (p) ι π | U π ) θ Y in the first and second commutative diagrams of Proposition 1.6, (i), and the commutative diagram of Remark 1.6.1 that we obtain relations ι λ g φ  c λ h τ c | U ) f Y = (p μ ) ξ g τ c | U ) f Y 1 2 c 2 2 c 2 = (p ) ι π p | U πp ) θ g f Y μ = (p μ ) (p) ι π | U π ) θ Y , where 70 the composite of the first and second equalities implies that, after possi- bly replacing K by a finite extension field of K [which in fact may be  Y ) K - taken to be unramified cf. Lemma 2.10, (i)], the tautological ( R def  Y ) K = R  Y O K, lifts to an valued point y R of Y n , where we write ( R K  ( R Y ) K -valued point y R of a certain intermediate covering W n Y n of W n Y n that corresponds to multiplication by p μ on the codomain of  m ), while the homomorphism f Hom O K ( T  qtr,n , G the third equality implies [cf. also the “essentially cartesian” nature of the squares in the commutative diagram of Remark 1.6.1] that y R lifts def  W ) K = R  W O K,  W ) K -valued point w R of W n , where ( R to an ( R K  and R W denotes the p-adic completion of some localization R W of R W/Y  W admits a at a maximal ideal of R W/Y such that the spectrum of R  tautological isomorphism over R Y to the spectrum of the p-adic completion of “Γ(O Y , Y )” at the generic point of “Y s [i.e., where the quotation marks refer to the notation of Proposition 1.6, (ii)]. In particular, since i 0 is not a [nonnegative integral] power of p, it follows from  W , hence also Proposition 1.6, (ii), and Remark 1.6.2 that the residue field of R the residue field of R W , is the function field of a curve over the residue field of O K of genus 1. Thus, we conclude from Proposition 2.3, (ii), (iii), that, after possibly replacing K by a finite extension field of K, there exist a compactified semistable model W n of W n over O K , together with a dominant morphism ψ : W n −→ Y n over O K , such that ψ restricts to the finite étale Galois covering ψ η : W n Y n ; R W is the local ring of W n at the generic point of an irreducible component D of (W n ) s whose normalization is of genus 1; ψ(D) = (y n ) s (Y n ) s . This completes the proof of Theorem 2.16. Theorem 2.17 (Resolution of nonsingularities for arbitrary hyperbolic curves over p-adic local fields). Let Σ Primes be a subset of cardinality 2; K a p-adic local field, for some p Σ; X a hyperbolic curve over K; L a mixed characteristic complete discrete valuation field of residue characteristic p that contains K as a topological subfield. Then X L satisfies Σ-RNS if and only if the residue field of L is algebraic over the finite field of cardinality p. Proof. First, we observe that it follows formally from Remark 2.2.3, (v), that it suffices to verify that X satisfies Σ-RNS. Next, we observe that it follows 71 immediately from the various definitions involved that we may assume without loss of generality that X has stable reduction over K. Write X for the [unique, up to unique isomorphism] compactified stable model of X over O K . Then, in light of Propositions 2.3, (xii); 2.4, (i), (ii), by replacing X by the [unique, up to unique isomorphism] smooth compactification of a suitable connected geometrically pro-Σ finite étale covering of X, we may assume without loss of generality that: X is a proper hyperbolic curve over K, X s is split, X s is singular, and every irreducible component of X s is a smooth curve of genus 2. In particular, X now satisfies the assumptions imposed in the respective initial portions of Proposition 2.6 and Definition 2.7. Next, observe that since the covering Y n Y is combinatorial [cf. Definition 2.7, (ii)], it follows immediately that this covering induces a surjection Y n (K)  Y (K) on K-rational points. Thus, it follows immediately from Proposition 2.4, (iii), and Theorem 2.16 that X satisfies Σ-RNS. This completes the proof of Theorem 2.17. 3 Point-theoreticity, metric-admissibility, and arith- metic cuspidalization Let p be a prime number. In the present section, we first recall the well- known classification of the points of the topological Berkovich space associated to a proper hyperbolic curve over a mixed characteristic complete discrete val- uation field via the notion of type i points, where i {1, 2, 3, 4} [cf. Defi- nition 3.1]. Next, we introduce a certain combinatorial classification of the VE-chains considered in §2 [cf. Definition 3.2] and observe that this classifica- tion of VE-chains leads naturally to a purely combinatorial characterization of the well-known classification via type i points mentioned above [cf. Proposi- tions 3.3, 3.4]. This combinatorial classification/characterization [cf. also the approach of Propositions 3.7, 3.8] was motivated by the argument applied in the proof of [CbTpIV], Theorem A.7. We then apply the theory of §2 to give a group-theoretic characterization, motivated by [but by no means identical to] the characterization of [Lpg2], §4, of the type i points in terms of the geometric Σ-tempered fundamental group of the hyperbolic curve [cf. Propositions 3.5, 3.9]. Then, by combining this group-theoretic characterization with [AbsTopII], Corollary 2.9, we prove an absolute version of the Grothendieck Conjecture for hyperbolic curves over p-adic local fields [cf. Theorem 3.12]. This settles one of the major open questions in anabelian geometry. As a corollary of this absolute version of the Grothendieck Conjecture for hyperbolic curves over p-adic local 72 fields, together with [HMM], Theorem A, we also obtain an absolute version of the Grothendieck Conjecture for configuration spaces associated to hyper- bolic curves over p-adic local fields [cf. Theorem 3.13]. We then switch gears to discuss metric-admissibility for p-adic hyperbolic curves. This discussion of metric-admissibility leads to a proof that all of the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature in fact coin- cide [cf. Theorem 3.16]. Moreover, as an application of Corollary 2.5, (i), and the theory developed in the present section, together with the theory of metric- admissibility developed in [CbTpIII], §3, we obtain a construction of a certain type of arithmetic cuspidalization of the [Primes-] tempered fundamental group of a hyperbolic curve over Q p [cf. Theorem 3.20]. Definition 3.1. Let Σ Primes be a nonempty subset; K a mixed charac- teristic complete discrete valuation field of residue characteristic p; X a proper  X a universal geometrically pro-Σ covering of X. hyperbolic curve over K; X Write Ω for the p-adic completion of [some fixed] K; (−) an for the topological  an [cf. Berkovich space associated to (−). Let x X an be an element; X Proposition 2.3, (vii), (viii)] a lifting of x. Then: (i) We shall say that is of type 1 if is determined by a point-theoretic  of X  associated to some point p-valuation on the function field K( X)  X(Ω) [cf. Definition 2.2, (ii)]. (ii) We shall say that is of type 2 if is determined by an inverse system of discrete residue-transcendental p-valuations associated to irreducible components of the special fibers of [compactified] semistable models with split reduction of the domain curves of connected finite étale coverings  Z X [cf. Proposition 2.3, Z X equipped with a factorization X (ii), (iii)]. (iii) We shall say that is of type 3 if there exist a finite extension field L of K, a [compactified] semistable model with split reduction X of X L over O L , and a node e of X s such that arises as the inverse image of a lifting  tor of some element D e VE(X ) tor [cf. Definition 2.2, (vi)] VE( X)  an  tor [cf. Proposition 2.3, (viii)]. via the homeomorphism X VE( X) (iv) We shall say that is of type 4 if, for each i {1, 2, 3}, is not of type i. (v) For each i {1, 2, 3, 4}, we shall say that x is of type i if is of type i. [One verifies immediately that, for each i {1, 2, 3, 4}, the condition that x is of type i is independent of the choices of Σ and x̃.] For each  an [i] X  an ) i {1, 2, 3, 4}, we shall write X an [i] X an (respectively, X  an ). for the subset of points of type i of X an (respectively, X 73 Remark 3.1.1. In the notation of Definition 3.1, we observe that  an = X  an [1] X  an [2] X  an [3] X  an [4]; X X an = X an [1] X an [2] X an [3] X an [4], and, moreover, for each pair of distinct i, j {1, 2, 3, 4},  an [j] = ∅;  an [i] X X X an [i] X an [j] = ∅.  an [j] = ∅. First, by considering  an [i] X Indeed, it suffices to verify that X the residue fields of the valuation rings under consideration, we conclude that  an [2] = ∅. Next, we observe that it follows immediately from the  an [1] X X discussion of the construction of “VE(Z) tor in Definition 2.2, (vi), that, for  an [i] X  an [3] = ∅. Finally, we observe that it is a tautology each i {1, 2}, X  an [i] X  an [4] = ∅. This completes the proof of that, for each i {1, 2, 3}, X relations in the above display. Definition 3.2. We maintain the notation of Definition 3.1. Let c = (c Z ) Z∈S  where S denotes the directed set of [compactified] semistable models VE( X),  [cf. Definition 2.2, (iii)]. Write V c S that appear in the definition of VE( X) (respectively, E c S) for the subset of [compactified] semistable models Z such that c Z is a vertex (respectively, an edge). Then: (i) We shall say that c is asymptotically verticial (respectively, asymptotically edge-like) if the subset V c S (respectively, E c S) forms a cofinal subset of S. [In particular, if c is asymptotically verticial (respectively, asymptotically edge-like), then V c (respectively, E c ) may be regarded as a directed set in a natural way.] (ii) Suppose that c is asymptotically verticial. Then we shall say that c is strongly verticial if there exists a cofinal subset S c V c satisfying the following condition: Let Z 1 , Z 2 S c be distinct elements such that Z 2 dominates Z 1 . Then the generic point of the irreducible component of (Z 2 ) s that corresponds to c Z 2 maps to the generic point of the irreducible component of (Z 1 ) s that corresponds to c Z 1 via the dominant morphism Z 2 Z 1 . (iii) Suppose that c is asymptotically verticial. Then we shall say that c is weakly verticial if there exists a cofinal subset S c V c satisfying the following condition: Let Z 1 , Z 2 S c be distinct elements such that Z 2 dominates Z 1 . Then the generic point of the irreducible component of (Z 2 ) s that corresponds to c Z 2 maps to a closed point in the interior of the irreducible component of (Z 1 ) s that corresponds to c Z 1 via the dominant morphism Z 2 Z 1 . 74 (iv) Suppose that c is asymptotically edge-like. Then we shall say that c is weakly edge-like if there exists a cofinal subset S c E c satisfying the following condition: Let Z 1 , Z 2 S c be distinct elements such that Z 2 dominates Z 1 . Then there exists a toral compactified semistable model Z 1 relative to Z 1 such that the dominant morphism Z 2 Z 1 admits a factorization Z 2 Z 1 Z 1 , and the node of (Z 2 ) s that corresponds to c Z 2 maps to a closed point in the interior of an irreducible component of (Z 1 ) s [that necessarily lies over the node of (Z 1 ) s that corresponds to c Z 1 ] via the dominant morphism Z 2 Z 1 . (v) Suppose that c is asymptotically edge-like. Then we shall say that c is strongly edge-like if there exists a cofinal subset S c E c satisfying the following condition: Let Z 1 , Z 2 S c be distinct elements such that Z 2 dominates Z 1 . Then, for each toral compactified semistable model Z 1 relative to Z 1 that admits a factorization Z 2 Z 1 Z 1 , the node of (Z 2 ) s that corresponds to c Z 2 maps to a node of (Z 1 ) s [that necessarily lies over the node of (Z 1 ) s that corresponds to c Z 1 ] via the dominant morphism Z 2 Z 1 . (vi) We shall write   vtc VE( X); VE( X)  edg VE( X);  VE( X)  str-vtc VE( X);  VE( X)  wk-vtc VE( X);  VE( X)  str-edg VE( X);  VE( X)  wk-edg VE( X),  VE( X) respectively, for the subsets of asymptotically verticial VE-chains, asymp- totically edge-like VE-chains, strongly verticial VE-chains, weakly verti- cial VE-chains, strongly edge-like VE-chains, and weakly edge-like VE- chains. Also, for each  {vtc, edg, str-vtc, wk-vtc, str-edg, wk-edg}, we shall write  prim, = VE( X)  prim VE( X)   VE( X) def  (⊆ VE( X)). (vii) Let Z S. Then in the notation of Definition 2.2, (vi), we shall write def VE(Z) tor,rat =  V w (⊆ VE(Z) tor ); D e (⊆ VE(Z) tor ). w∈V(Z) def VE(Z) tor,irr =  e∈E(Z) 75 Note that one verifies immediately that VE(Z) tor = VE(Z) tor,rat  VE(Z) tor,irr , and that, for any Z 1 , Z 2 S such that Z 2 dominates Z 1 , the natural map VE(Z 2 ) tor VE(Z 1 ) tor induces a map VE(Z 2 ) tor,rat VE(Z 1 ) tor,rat [cf. the discussion of Definition 2.2, (vi)]. Remark 3.2.1. In the notation of Definition 3.2, we observe that it follows im- mediately from the various definitions involved [cf. also Remarks 2.1.4, 2.1.5; Definition 2.2, (vi)] that:  str-vtc VE( X)  wk-vtc = ∅; VE( X)  str-edg VE( X)  wk-edg = ∅; VE( X)  vtc = VE( X)  str-vtc VE( X)  wk-vtc ; VE( X)  edg = VE( X)  str-edg VE( X)  wk-edg ; VE( X)  = VE( X)  vtc VE( X)  edg . VE( X) Proposition 3.3 (Elementary properties of the combinatorial classifi- cation of VE-chains). In the notation of Definition 3.2, by forming the in- ductive limit of the natural log structures on the [compactified] semistable models Z S, we obtain an ind-log structure on the pro-scheme lim Z∈S Z. Then the ←− following hold:  (i) Let c VE( X). Write c for the center on lim Z∈S Z of the valuation ←− ring R c associated to c [cf. Proposition 2.3, (vi)]; M c pf for the perfection of the inductive limit monoid obtained by forming the stalk at c of the characteristic of the ind-log structure on the pro-scheme lim Z∈S Z. Then, ←−  str-edg ), then M pf is isomorphic  vtc (respectively, c VE( X) if c VE( X) c to Q ≥0 (respectively, Q ≥0 × Q ≥0 ). (ii) The following relations hold:  str-vtc VE( X)  str-edg = ∅; VE( X)  str-vtc VE( X)  wk-vtc = ∅; VE( X)  wk-vtc VE( X)  str-edg = ∅; VE( X)  wk-edg VE( X)  wk-vtc ; VE( X)  = VE( X)  str-vtc VE( X)  wk-vtc VE( X)  str-edg . VE( X) 76 Proof. Assertion (i) follows immediately from the well-known log structure of M c pf [cf. also the discussion of the subsection in Notations and Conventions en- titled “Log schemes”], together with the various definitions involved. Assertion (ii) follows immediately from assertion (i), together with the various defini- tions involved [cf. also Remark 3.2.1]. This completes the proof of Proposition 3.3. Proposition 3.4 (Characterization of points of type 2 and 3 via the combinatorial classification of VE-chains). We maintain the notation of Definition 3.2. Then the following hold:  \ VE( X)  prim . Then  nonprim def = VE( X) (i) Write VE( X)  nonprim VE( X)  prim,str-edg .  str-edg = VE( X) VE( X) Moreover, the unique nontrivial generization of a nonprimitive VE-chain [cf. Proposition 2.3, (x)] is strongly verticial. (ii) For Z S, write τ X  str-edg  str-edg VE( X)  −→  tor −→ VE(Z) tor : VE( X) VE( X) τ X,Z  for the natural composite map [cf. Proposition 2.3, (viii)]. Then for any str-edg induces a map Z S, τ X,Z   nonprim VE(Z) tor,rat VE( X)  prim,str-edg , there exists an element [cf. (i)]. Moreover, for each c VE( X) Z c S such that for every Z S that dominates Z c , str-edg (c) VE(Z) tor,irr τ X,Z  [cf. (i)]. Finally, Z c S may be taken to be a [compactified] semistable model with split reduction of X L over O L for some finite extension field L of K.  an  prim X [cf. Proposition 2.3, (viii)] determines (iii) The bijection VE( X) bijections  an [2];  prim,str-vtc X VE( X)  prim,str-edg X  an [3]. VE( X) (iv) Let Y be a proper hyperbolic curve over K; f : Y X a dominant def morphism over K; y Y an . Write x = f (y) X an . Then, for each i {1, 2, 3, 4}, y is of type i if and only if x is of type i. 77 Proof. Assertion (i) follows immediately from the various definitions involved.  str-edg . First, suppose that Next, we verify assertion (ii). Let c VE( X) nonprim  prim,str-vtc   c VE( X) . Write c VE( X) for the unique nontrivial gener- ization of c [cf. assertion (i)]. Then it follows immediately from the definition of τ X  in the proof of Proposition 2.3, (viii), that τ X  (c) = τ X  (c  ). In particular, it  prim,str-vtc that τ  (c  ) maps to follows immediately from the fact that c  VE( X) X str-edg (c) VE(Z) tor,rat , an element of VE(Z) tor,rat for any Z S, hence that τ X,Z   prim,str-edg . Let Z c S c for some as desired. Next, suppose that c VE( X) “S c as in Definition 3.2, (v). Let Z S be an element that dominates Z c . Then one verifies immediately [cf. also the final portion of Definition 3.2, (vii)] str-edg (c) VE(Z) tor,rat implies a contradiction to the condi- that any relation τ X  str-edg tion of Definition 3.2, (v). Thus, we conclude that τ X,Z (c) VE(Z) tor,irr , as  desired. The fact that Z c S may be taken to be a [compactified] semistable model with split reduction of X L over O L for some finite extension field L of K follows immediately from Proposition 2.3, (iii), (iv). This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii), together with the various definitions involved [cf. also the final portion of Definition 3.2, (vii); Proposition 3.3, (ii)]. Finally, we consider assertion (iv). First, we observe that the asserted equivalence follows immediately from the various definitions involved [cf. also Proposition 2.3, (ii), (iii), in the case where i = 2] when i {1, 2}. Thus, it suffices to verify the asserted equivalence when i = 3. When i = 3, sufficiency follows immediately, in light of assertion (iii) and Proposition 3.3, (ii), from Definition 3.1, (iii) [cf. also the discussion of Definition 2.2, (vi)]. On the other hand, it follows immediately, by replacing Y by the normalization of Y in the Galois closure of the finite extension of function fields determined by f and applying the sufficiency that has already been verified, that to verify necessity when i = 3, we may assume without loss of generality that the finite extension of function fields determined by f is Galois. But then the desired necessity follows immediately from the final portion of assertion (ii), together with Proposition 2.3, (iii), (iv) [cf. also the discussion of Definition 2.2, (vi)]. This completes the proof of Proposition 3.4. Proposition 3.5 (Types of points and geometrically pro-l decomposi- tion groups). In the notation of Definition 3.2, let l Σ \ {p}. Suppose that X satisfies Σ-RNS. Then the following hold: (i) Let  wk-vtc ; c VE( X)  str-edg ).  str-vtc (respectively, c VE( X) c VE( X)  →) Z For each connected geometrically pro-Σ finite étale covering ( X def {l} X, write D Z,c Δ lZ = Δ Z for the decomposition subgroup associated to c of the geometric pro-l fundamental group of Z [cf. the subsection in 78 Notations and Conventions entitled “Fundamental groups”]. Then there  →) Y X exists a connected geometrically pro-Σ finite étale covering ( X of X such that, for each connected geometrically pro-Σ finite étale covering  →) Z Y of Y , D Z,c is isomorphic to a [nonabelian] pro-l surface ( X group [cf. [MT], Definition 1.2] (respectively, the trivial group; Z l ).  an and the subset X  an [i] X  an may (ii) Let i {2, 3}. Then the set X be reconstructed, functorially with respect to isomorphisms of topological groups, from the underlying topological group of the geometric Σ-tempered fundamental group of X [cf. the subsection in Notations and Conventions entitled “Fundamental groups”]. (iii) Let Y , Z be [not necessarily proper!] hyperbolic curves over K that satisfy Σ-RNS; Y  Y , Z  Z universal geometrically pro-Σ coverings of Y , Z, respectively; f : Y Z a dominant morphism over K; H G K a closed subgroup such that the restriction to H of the l-adic cyclotomic charac- ter of K has open image, and, moreover, the intersection H ∩I K of H with the inertia subgroup I K of G K admits a surjection to [the profinite group] (Σ) def (Σ) def  = Gal( Y  /Y ), s Z : H Π = Gal( X/X) sections Z l ; s Y : H Π Y Z (Σ) of the restrictions to H of the respective natural surjections Π Y  G K , (Σ) (Σ) Π Z  G K such that s Y is mapped, up to Π Z -conjugation, by f to s Z via the map induced by f on geometrically pro-Σ fundamental groups. Write Ω H Ω for the subfield of Ω fixed by H. Then s Y arises from a(n) [necessarily unique] Ω H -rational point Y H ) if and only if s Z arises from a(n) [necessarily unique] Ω H -rational point Z(Ω H ). Proof. Since X satisfies Σ-RNS [cf. Definition 2.2, (vii)], assertion (i) follows immediately from the well-known structure of the maximal pro-l quotient of the admissible fundamental group of a stable curve over a separably closed field of characteristic p [cf. e.g., [SemiAn], Example 2.10], together with the various def- initions involved. Assertion (ii) follows immediately from assertion (i), together with Corollary 2.5, (i); Proposition 3.4, (iii) [cf. also Proposition 3.3, (ii)]. Fi- nally, we consider assertion (iii). First, we observe that it follows immediately (Σ) from the profinite nature of the topological group Π Y that, by replacing Y and Z by the smooth compactifications of the various finite étale coverings of Y and Z corresponding, respectively, to suitable open neighborhoods of the image of (Σ) (Σ) s Y in Π Y × G K H and the image of s Z in Π Z × G K H, we may assume without loss of generality that Y and Z are proper. Next, we observe that the various uniqueness assertions in the statement of Proposition 3.5, (iii), follow immedi- ately from the final portion of Proposition 2.4, (vii), and that necessity follows immediately from the various definitions involved. Thus, it suffices to verify sufficiency. On the other hand, sufficiency follows, in light of the equivalences of Proposition 3.4, (iv), formally from the final portion of Proposition 2.4, (vii). This completes the proof of Proposition 3.5. 79 Definition 3.6. Let K be a mixed characteristic complete discrete valuation field of residue characteristic p; X a proper hyperbolic curve over K. Write K(X K ) for the function field of X K . Let v be a p-valuation on K(X K ). Write (O K ⊆) O v K(X K ) for the valuation ring associated to v. [Note that it follows immediately from the well-known theory of one-dimensional function def fields that (O v ) K = O v · K K(X K ) is equal either to K(X K ) or to the discrete valuation ring associated to a closed point of X K .] Then: (i) Let M be an O v -module. Then we shall say that M is bounded if the image def of M via the natural morphism M M K = M O K K is contained in a finitely generated O v -submodule of M K . We shall say that M is unbounded if M is not bounded. (ii) We shall say that the p-valuation v is differentially bounded (respectively, differentially unbounded) if the O v -module of relative differentials Ω O v /O K is bounded (respectively, unbounded). Proposition 3.7 (Approximation of closed points of the generic fiber via generic points of special fibers). Let K be a mixed characteristic com- plete discrete valuation field of residue characteristic p; K = K 0 K 1 · · · K i · · · an ascending chain of finite extension fields of K contained in K and indexed by N. Write Ω for the p-adic completion of K. Let X be a hyperbolic curve over K. For each i N, let X i be a compactified semistable model with split reduction of X K i over O K i ; φ i+1 : X i+1 −→ X i × O Ki O K i+1 a dominant morphism over O K i+1 that induces the identity automorphism on the generic fiber; v i an irreducible component of (X i ) s . Suppose that, for each i N, the projection to X i of φ i+1 (v i+1 ) is a closed point x i (X i ) s X i of (X i ) s that lies in the smooth locus of v i . Then the following hold: (i) For each i N, let ψ i : Spec O K i → X i be a section whose image contains def x i . Then there exists a collection {t i , γ i+1 , π i+1 } i∈N of elements t i A i = O X i ,x i and γ i+1 , π i+1 m K i+1 such that, for each i N, t i A i is a generator of the ideal that defines the scheme-theoretic image of ψ i , and t i = γ i+1 + π i+1 t i+1 , where we regard A i as a subring of A i+1 via the injection A i → A i+1 induced by the composite X i+1 X i [which maps x i+1 → x i ] of φ i+1 with the projection to X i . 80 def (ii) We maintain the notation of (i). For each positive integer i, write l i = v p i ). Suppose that the equality  l i = +∞ i≥1 holds. Then there exists a closed point x Ω of X Ω such that, for each i N, the center on X i of the closed point x Ω of X Ω , hence also of the point-theoretic p-valuation on the function field of X K i determined by the closed point x Ω of X Ω , coincides with x i . Proof. First, we verify assertion (i). We construct elements t i A i and γ i+1 , π i+1 m K i+1 , for i N, by induction on i N. Let t 0 A 0 be a generator of the ideal that defines the scheme-theoretic image of ψ 0 ; i N. Suppose that the elements t j+1 , γ j+1 , and π j+1 have been constructed for j N such that j < i. Write γ i+1 m K i+1 for the image of t i via the composite homomor- phism A i A i+1 O K i+1 induced by ψ i+1 . Next, observe that the image in A i+1 O Ki+1 K i+1 of t i γ i+1 is a generator of the maximal ideal associated to the closed point of X K i+1 determined by ψ i+1 . Thus, since x i+1 lies in the smooth locus of (X i+1 ) s , and A i+1 is a regular local ring, hence a unique fac- torization domain, we conclude that there exists an element π i+1 m K i+1 such that t i −γ i+1 = π i+1 t i+1 for some generator t i+1 A i+1 of the ideal that defines the scheme-theoretic image of ψ i+1 . This completes  the proof of assertion (i). Next, we verify assertion (ii). Suppose that i≥1 l i = +∞. For each i N, fix elements t i A i and γ i+1 , π i+1 m K i+1 as in the statement of assertion (i). For each positive integer j, write def s j = γ j · π i . 1≤i≤j−1 it follows immediately from our assump- Then since  O Ω is p-adically complete,  tion that i≥1 l i = +∞ that j≥1 s j converges to an element γ m Ω . Write x Ω for the closed point of X Ω determined by the homomorphism ψ Ω : A 0 O Ω over O K 0 that maps t 0 → γ. Now observe that it follows immediately from the definition of γ that, for each i N, ψ Ω extends uniquely to a homomorphism A i O Ω over O K i . On the other hand, the existence of such unique exten- sions implies that the center on X i of the closed point x Ω of X Ω , hence also of the point-theoretic p-valuation on the function field of X K i determined by the closed point x Ω of X Ω , coincides with x i . This completes the proof of assertion (ii), hence of Proposition 3.7. Proposition 3.8 (Characterization of points of type 1 via differentially unboundedness). In the notation of Definition 3.6 [cf. also the notation of  (⊇ K(X )) that restricts Proposition 2.3, (viii)], let v  be a p-valuation of K( X) K to v on K(X K ) [cf. Remark 2.2.4]. Suppose that v  is primitive. Then the point  an associated to v  [cf. Proposition 2.3, (viii)] is of type 1 if and only if x v  X v is differentially unbounded. 81  an . Note that, in Proof. Write x v X an for the point determined by x v  X light of Remark 3.1.1, it suffices to verify that if x v is of type 1 (respectively, of type i {2, 3, 4}), then v is differentially unbounded (respectively, differentially bounded). First, we verify that if x v is of type 1, then v is differentially unbounded. Suppose that x v is of type 1. Write Ω for the p-adic completion of K; e v  : O v O Ω for the natural evaluation homomorphism over O K associated to x v  . Next, observe that, since O v O K K is a valuation ring [contained in the function field K(X K ) of the hyperbolic curve X K over K] that contains K, and whose field of fractions coincides with K(X K ) [cf. Definition 2.2, (ii)], it follows immediately from the well-known theory of one-dimensional function fields over an algebraically closed field that Ω O v /O K O K K is a rank one free module over O v O K K. In particular, there exists an element t O v such that e v  (t) m Ω , and dt is a free generator of Ω O v /O K O K K over O v O K K. Next, we observe that, for any positive integer N , there exists an element a N m K such that e v  (t a N ) = e v  (t) a N p N O Ω . N O v . On the other hand, it Note that the above equation implies that t−a p N follows immediately from the definition of the module of relative differentials that N dt = d(t a N ) = p N · d( t−a ) Ω O v /O K . p N In particular, we conclude that dt is a nonzero p-divisible element of Ω O v /O K . Thus, since dt is a free generator of Ω O v /O K O K K over O v O K K, the dif- ferential boundedness of v would imply that arbitrary negative integral powers of p are contained in O v , i.e., in contradiction to our assumption that v is a p-valuation. Hence we conclude that v is differentially unbounded, as desired. Next, we verify that, if x v is of type 2, then v is differentially bounded. Suppose that x v is of type 2. Then it follows immediately from Proposition 3.4, (iii), that there exist a finite extension field L of K, a compactified semistable model X of X L over O L , and a generic point x of X s such that O v O X ,x O L O K . In particular, to verify that v is differentially bounded, it suffices to verify that Ω O X ,x /O L is a finitely generated O X ,x -module. On the other hand, this follows immediately from the fact that O X ,x is essentially of finite type over O L . Next, we verify that, if x v is of type 3, then v is differentially bounded. Suppose that x v is of type 3. Then it follows from Proposition 3.4, (iii), that the VE-chain associated to the p-valuation v  is strongly edge-like. Thus, it follows immediately from Proposition 2.3, (iii), (iv) [cf. also Definition 3.2, (v), [the final portion of] (vii); Proposition 3.4, (ii)] that there exist an ascending chain K = K 0 K 1 · · · K i · · · of finite extension fields of K contained in K and indexed by N 82 and, for each i N, a compactified semistable model X i with split reduction of X K i over O K i , a dominant morphism φ i+1 : X i+1 −→ X i × O Ki O K i+1 over O K i+1 that induces the identity automorphism on the generic fiber, a node e i of (X i ) s satisfying the following conditions: For each i N, the projection to X i of φ i+1 (e i+1 ) coincides with the node e i (X i ) s X i . The equality O v = lim A i O Ki O K −→ i∈N def where, for each i N, we write A i = O X i ,e i ; the transition map is the homomorphism A i O Ki O K A i+1 O Ki+1 O K induced by the composite X i+1 X i [which maps e i+1 → e i ] of φ i+1 with the projection to X i holds [cf. Proposition 2.3, (vi)]. def For each i N, write S i = Spec O K i ; S i log for the log scheme determined by the log structure on S i associated to the closed point of S i ; X i log for the log scheme over S i log determined by the natural log structure on X i [i.e., the multiplicative monoid of sections of O X i that are invertible on the open subscheme of X i determined by X K i ]; ω X log /S log ,e i for the stalk at e i of the sheaf of relative i i logarithmic differentials associated to the proper, log smooth morphism X i log S i log [cf. the subsection in Notations and Conventions entitled “Log schemes”]. Then it follows immediately from the definitions of the various log structures involved that the morphism φ i+1 : X i+1 X i × O Ki O K i+1 = X i × S i S i+1 extends to a log étale morphism of log schemes log log −→ X i log × S log S i+1 , X i+1 i which induces a natural isomorphism ω X log /S log ,e i A i A i+1 ω X log /S log ,e i+1 i i i+1 i+1 of A i+1 -modules, hence a natural homomorphism φ : Ω O v /O K = lim Ω A i O K O K /O K −→ ω X log /S log ,e 0 A 0 O v −→ 0 0 i i∈N 83 of O v -modules. Here, we note that φ induces a natural isomorphism Ω O v /O K O K K ω X log /S log ,e 0 A 0 O v O K K 0 0 of free O v O K K-modules of rank 1. Thus, since ω X log /S log ,e 0 is a finitely 0 0 generated A 0 -module, we conclude that v is differentially bounded, as desired. Finally, we verify that, if x v is of type 4, then v is differentially bounded. Suppose that x v is of type 4. Then it follows from Proposition 3.3, (ii); Propo- sition 3.4, (iii) [cf. also Remark 3.1.1], that the VE-chain associated to the p-valuation v  [cf. Proposition 2.3, (viii)] is weakly verticial. Thus, it follows immediately from Proposition 2.3, (iii), (iv) [cf. also Definition 3.2, (iii); the final portion of Remark 2.1.4] that there exist an ascending chain K = K 0 K 1 · · · K i · · · of finite extension fields of K contained in K and indexed by N and, for each i N, a compactified semistable model X i with split reduction of X K i over O K i , a dominant morphism φ i+1 : X i+1 −→ X i × O Ki O K i+1 over O K i+1 that induces the identity automorphism on the generic fiber, an irreducible component v i of (X i ) s satisfying the following conditions: For each i N, the projection to X i of φ i+1 (v i+1 ) is a closed point x i (X i ) s X i of (X i ) s that lies in the smooth locus of v i . For each i N, there exists a section ψ i : Spec O K i → X i whose image contains x i . The equality O v = lim A i O Ki O K −→ i∈N def where, for each i N, we write A i = O X i ,x i ; the transition map is the homomorphism A i O Ki O K A i+1 O Ki+1 O K induced by the composite X i+1 X i [which maps x i+1 → x i ] of φ i+1 with the projection to X i holds [cf. Proposition 2.3, (vi)]. 84 Thus, we are in the situation of Proposition 3.7, (i). In particular, there exists a collection {t i , γ i+1 , π i+1 } i∈N of elements t i A i and γ i+1 , π i+1 m K i+1 as in Proposition 3.7, (i), such that t i = γ i+1 + π i+1 t i+1 . Next, observe that since x i lies in the smooth locus of (X i ) s , it follows that Ω A i /O Ki is a free A i -module of rank 1 generated by dt i . In particular, since t i = γ i+1 + π i+1 t i+1 , we conclude that 1 Ω A i+1 /O Ki+1 = · Ω A i /O Ki A i A i+1 . π i+1 Thus, it follows immediately from the equality O v = lim i∈N A i O Ki O K that −→ 1 · Ω A 0 /O K 0 A 0 O v . Ω O v /O K = lim  −→ 1≤j≤i+1 π j i∈N Now suppose that v is differentially unbounded. For each positive integer i, def write l i = v p i ). Then since v is differentially unbounded, we conclude that the equality  l i = +∞ i≥1 holds. Next, we observe that it follows immediately from Proposition 3.7, (ii) [cf. also the equality O v = lim i∈N A i O Ki O K ], that v determines a closed −→ point x Ω of X Ω such that the valuation ring of the point-theoretic p-valuation on K(X K ) associated to x Ω dominates O v , hence coincides with O v . Thus, we conclude that x v is the point X an determined by x Ω , hence that x v is of type 1, in contradiction to our assumption that x v is of type 4. This completes the proof of Proposition 3.8. Proposition 3.9 (Characterization of points of type 1 via geometric Σ-tempered decomposition groups). In the notation of Definition 3.2, sup- pose that p Σ. Let x X an be an element; l Σ \ {p}; D x a decomposition of X associated to group in the geometric Σ-tempered fundamental group Δ Σ-tp X x. Then the following hold: (i) Suppose that x is of type 1. Then D x is trivial. (ii) Suppose that x is of type 4. Then there exists an open subgroup of D x that admits a continuous surjective homomorphism to Z p . In particular, D x is nontrivial. (iii) Suppose that x is of type i {2, 3}, and that X satisfies Σ-RNS. Then there exists an open subgroup of D x that admits a continuous surjective homomorphism to Z l . In particular, D x is nontrivial. (iv) Suppose that X satisfies Σ-RNS. Then x is of type 1 if and only if D x is trivial. 85  an be a lifting of x. First, we observe that assertion (i) follows Proof. Let x  X immediately from the various definitions involved. Next, we verify assertion (iii). Suppose that x [or, equivalently, x  ] is of type i {2, 3}, and that X satisfies Σ-RNS. Then it follows immediately from Proposition 3.4, (iii); Proposition 3.5, (i), that there exists an open subgroup of D x that admits a continuous surjective homomorphism to Z l . This completes the proof of assertion (iii). Assertion (iv) follows immediately from assertions (i), (ii), (iii) [cf. also Remark 3.1.1]. Thus, to complete the proof of Proposition 3.9, it suffices to verify assertion (ii). To verify assertion (ii), by replacing K by a suitable extension field of K contained in Ω, we may assume without loss of generality [cf. Proposition 3.4, (iii)] that the residue field of K is separably closed. Suppose that x [or, equivalently, x  ] is of type 4, and that no open subgroup of D x admits a continuous  surjective homomorphism to Z p . Write v  for the primitive p-valuation on K( X) associated to x  [cf. Proposition 2.3, (viii)]; v for the p-valuation obtained by restricting v  to K(X K ). Then since x is of type 4, it follows from Proposition 3.8 that Ω O v /O K is bounded. In particular, by replacing K by a finite extension field of K, if necessary, we observe that there exist a positive integer N and a compactified semistable model with split reduction X of X over O K such that the center z on X of the VE-chain associated to v  lies in the smooth locus of X s X , arises from a point of X valued in the residue field of O K , and satisfies the following condition [cf. the portion of the proof of Proposition 3.8 concerning points of type 4]: O X ,z /O K ⊆) Ω O X ,z /O K O X ,z O v Ω O v /O K 1 · Ω O X ,z /O K O X ,z O v p N where by a slight abuse of notation, we use the notation “⊆” to denote the various natural inclusions, and we note that since Ω O X ,z /O K is a free O X ,z - module of rank 1, the O v -module Ω O X ,z /O K O X ,z O v is a free O v -module of rank 1. In particular, it follows immediately that the second and third inclusions of the above display induce the injections on the respective p-adic completions [cf. the discussion of Remark 2.2.4]. In the remainder of the proof of assertion (ii), we suppose that we are in the situation of Proposition 2.12. Moreover, by replacing X by a suitable geo- metrically pro-Σ connected finite étale covering of X [cf. Proposition 2.3, (xii); Definition 2.7], we may assume without loss of generality that X = Y n , X = Y n , z = (y n ) s ,  X ,z = O  Y ,y , O n n  X ,z denotes the completion of the local ring O X ,z . Write where O  m ) −→ O  × Ψ : Hom O K ( T  qtr,n , G X ,z  m ) for the assignment discussed in Proposition 2.13, (ii). Let f Hom O K ( T  qtr,n , G be a nontrivial element [which exists by Propositions 2.11, (i); 2.13, (i)]. Thus, 86 the logarithmic differential def def ) θ = dΨ(f  X ,z /O K = lim Ω (O X ,z /m m Ψ(f ) Ω O z )/O K ←− m≥1 where m z denotes the maximal ideal of O X ,z ; m ranges over the positive  × is = 0 [cf. the first display, as well as the discussion integers of Ψ(f ) O X ,z following this first display, in the proof of Theorem 2.16, where we observe that this portion of the proof of Theorem 2.16 may be applied even in the case of the “K” i.e., with separably closed residue field of the present discussion].  v for the field of fractions of  v for the p-adic completion of O v ; K Next, write O  v . Since v is a real valuation [cf. Proposition 2.3, (vii); Remark 3.1.1], it follows O immediately from the final portion of Remark 2.2.4 that the henselization of O v  v . Write H Δ Σ-tp for the closed subgroup may be regarded as a subring of O X obtained by forming the intersection of the kernels of the continuous surjective  H , K( X)  D x K( X)  for the subfields fixed homomorphisms Δ Σ-tp  Z p ; K( X) X by H and D x , respectively. Then since there does not exist any continuous surjective homomorphism D x  Z p , we thus conclude that D x H, hence that  H K( X)  D x K  v . K( X) On the other hand, it follows immediately from the definition of the center z on X that there exists a natural homomorphism φ : O X ,z O v of local rings,  X ,z O  v of topological local rings. which thus induces a homomorphism φ  : O def  = 0. Now we claim that φ  is injective. Indeed, suppose that p = Ker( φ)   Then since O X ,z is a regular local ring of dimension 2, and O v is p-torsion-free, it follows that p is a prime ideal of height 1 such that p O K = {0}. Next,  X ,z /p)⊗ O (O K /m K ) is finite over O K /m K . Thus, since O  X ,z /p observe that ( O K  is a complete O K -module, we conclude that O X ,z /p is finite over O K . On the  v is other hand, observe that the composite homomorphism O X ,z → O v → O  X ,z /p is injective. injective, hence that the natural homomorphism O X ,z O In particular, we conclude that O X ,z O K K embeds into a finite dimensional K-vector space, a contradiction. This completes the proof of our claim that φ  is injective. Next, we observe that since the image of Ψ(f ) is p-divisible in the multi-  X ,z O  H } × [cf. Proposition 2.13, (ii)], the image plicative group { O K( X) X ,z  × , hence also in the multiplicative group of Ψ(f ) in the multiplicative group K v ×  O v , is p-divisible. Write Ω O  v /O def K = lim Ω (O v /p m ·O v )/O K , ←− m≥1 where m ranges over the positive integers. Then it follows immediately from well-known basic facts concerning modules of differentials, together with the fact that Ω O X ,z /O K is a free O X ,z -module of rank 1, that  Ω O  X ,z /O K = lim Ω O X ,z /O K O X ,z O X ,z /m m z = Ω O X ,z /O K O X ,z O X ,z ; ←− m≥1 87 Ω O  v /O K = lim Ω O v /O K Z Z/p m Z. ←− m≥1 In particular, since φ  is injective, it thus follows from the discussion of the final portion of the second paragraph of the present proof that  X ,z Ω O /O O O  v Ω  Ω O  X ,z /O K = Ω O X ,z /O K O X ,z O X ,z X ,z K O v /O , K hence that the image θ v Ω O  v /O of θ in Ω O  v /O is = 0. On the other hand, K K  × is p-divisible, and Ω  since the image of Ψ(f ) in O is, by definition, p- O v /O K v adically separated, we conclude that θ v = 0, a contradiction. This completes the proof of assertion (ii), hence of Proposition 3.9. Corollary 3.10 (Reconstruction of points of type 1 via geometric tem- pered fundamental groups). Let Σ Primes be a subset of cardinality 2 that contains p; K a mixed characteristic complete discrete valuation field of residue characteristic p; X a hyperbolic curve over K. Write Ω for the p-adic  completion of K; Π tp (−) for the Σ-tempered fundamental group of (−); X X tp  for the universal pro-Σ covering corresponding to Π [so Gal( X/X) may be X identified with the pro-Σ completion of Π tp X ]. Suppose that X satisfies Σ-RNS.   Then the set X(Ω) equipped with its natural action by Gal( X/X) hence also,   the quotient set X(Ω)  X(Ω) by passing to the set of Gal( X/X)-orbits, may be reconstructed, in a purely combinatorial/group-theoretic way and func- torially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp X . Proof. Recall that, for any hyperbolic curve Y over K, the set of cuspidal inertia subgroups of Π tp Y , hence also the genus of Y , may be reconstructed, in a purely combinatorial/group-theoretic way and functorially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp Y [cf. the generalized version of [SemiAn], Corollary 3.11, discussed in [AbsTopII], Remark 2.11.1, (i)]. On the other hand, in the case where X is a proper hyperbolic curve over K, we observe that Corollary 3.10 follows immediately from Proposition 3.9, (iv), and [the proof of] Corollary 2.5, (i). Thus, by applying this observation to the Σ-tempered fundamental groups of the smooth compactifications of the various [connected] geometrically pro-Σ finite étale Galois coverings of X over K of genus 2, we conclude that   X(Ω) equipped with its natural action by Gal( X/X) may be reconstructed, in a purely combinatorial/group-theoretic way and functorially with respect to isomorphisms of topological groups, from the underlying topological group of Π tp X . This completes the proof of Corollary 3.10. 88 Theorem 3.11 (Preservation of decomposition subgroups associated to closed points). For  {†, ‡}, let p  be a prime number; Σ  Primes a subset that contains p  ; l \{p }) \{p }); K  a mixed characteristic complete discrete valuation fields of residue characteristic p  ; X  a hyperbolic  curve over K  ; L  K a tamely ramified [not necessarily finite!] Galois extension of K  that may be written as a union of finite tamely ramified Galois  extensions of K  in K of ramification index prime to l. Let σ ) ) Π X X L L be an isomorphism of profinite groups between the geometrically pro-Σ étale fundamental group of X L and the geometrically pro-Σ étale fundamental group of X L . For  {†, ‡}, write Δ Σ for the geometric pro-Σ  étale fundamental X   L  group of X L   , I L  G L  for the inertia subgroup of G L  , and k L  for the residue field of L  . Then the following hold: (i) We have an equality p = p , and σ induces isomorphisms of profinite Δ Σ , G L G L . In particular, Σ = Σ . Finally, if, groups Δ Σ X X L L for each  {†, ‡}, every pro-l closed subgroup of the kernel of the l-adic cyclotomic character on G k L  is trivial [cf. Remark 3.11.1 below], then, for all sufficiently small open subgroups J G L , J G L such that σ induces an isomorphism J J , σ also induces an isomorphism of profinite groups between the respective images of J I L , J I L in the maximal pro-l quotients J  (J ) {l} , J  (J ) {l} . (ii) Suppose that, for all sufficiently small open subgroups J G L , J G L such that σ induces an isomorphism J J , σ also induces an isomorphism of profinite groups between the respective images of J I L , J I L in the maximal pro-l quotients J  (J ) {l} , J  (J ) {l} . def Write Σ = Σ = Σ [cf. (i)]. Suppose, moreover, that X and X satisfy Σ-RNS. Then σ induces a bijection between the respective sets of decomposition subgroups associated to closed points of X L  and X L  , where  denote the respective completions of L , L .  , L L def def Proof. First, we verify assertion (i). Write τ = σ −1 , τ = σ. For  {†, ‡}, write   for the unique element of {†, ‡} \ {}. Then observe that it follows   immediately, by applying to τ  Σ  ), for both  = and  = ‡, X  L   the argument of the proof of [MiTs1], Corollary 4.6 [in the case where the extension L  /K  is finite; here, we note that in this case, it follows from  ) [MiSaTs], Theorem 3.8, that if Π X  is topologically finitely generated,  L   then the extension L  /K  is also finite], and 89 [MiSaTs], Theorem 3.8 [in the case where the extension L  /K  is infinite], that σ induces isomorphisms of profinite groups Σ Δ Σ X Δ X ; G L G L . Thus, we conclude from [MiTs2], Theorem A, (i), together with the well-known structure of geometric fundamental groups of hyperbolic curves over fields of characteristic zero [cf., e.g., [MT], Remark 1.2.2], that p = p , and Σ = Σ . The final portion of assertion (i) follows immediately from the well-known structure, for  {†, ‡}, of the Galois group Gal((K  ) tm /K  ) over K  of the maximal tamely ramified extension (K  ) tm of K  [under the assumption that every pro-l closed subgroup of the kernel of the l-adic cyclotomic character on G k L  is trivial], which implies that, for any sufficiently small open subgroup J  G L  , the image of J  I L  in the maximal pro-l quotient J   (J  ) {l} coincides with the unique maximal abelian normal closed subgroup of (J  ) {l} . This completes the proof of assertion (i). Next, we verify assertion (ii). First, we note that it follows from assertion def (i) that p = p = p . Thus, in light of our assumption on σ [cf. also assertion (i)], it follows from [CmbGC], Corollary 2.7, (i) [applied in the case where “l” is taken to be the p of the present discussion], (iii) [applied in the case where “l” is taken to be the l of the present discussion], that the isomorphism Δ Σ Δ Σ X X L L [cf. (i)] satisfies the condition (b ) of [CbTpIII], Proposition 3.6. In particular, by applying [CbTpIII], Proposition 3.6, (i), we conclude that the isomorphism Δ Σ Δ Σ arises, up to composition with an inner automorphism, from an X X L L isomorphism between the respective geometric Σ-tempered fundamental groups of X and X . Thus, by replacing σ by the composite of σ with an inner automorphism arising from Δ Σ , we may assume without loss of generality X L that σ arises from an isomorphism between the respective pull-backs via the natural inclusions G L G K , G L G K of the geometrically Σ-tempered fundamental groups of X and X .  , X  for the universal geometrically pro-Σ coverings corre- Next, write X (Σ) (Σ) sponding to Π , Π , respectively; Ω , Ω for the p-adic completions of X L X L K , K , respectively. Then since σ determines an isomorphism between the respective geometric Σ-tempered fundamental groups of X and X , it follows immediately from Corollary 3.10 that σ induces a bijection   ) X ) X that is compatible with the respective natural actions of Π (Σ) (Σ) , Π . X X L L Thus, in light of [Tate], §3.3, Theorem 1 [which, as is easily verified, admits a routine  ,  , L generalization to mixed characteristic complete valuation fields such as L i.e., whose valuations are not necessarily discrete [but nonetheless tamely ram- ified over some discrete valuation], and whose residue fields are not necessarily 90 perfect], we conclude that σ induces a bijection between the respective sets of decomposition subgroups associated to closed points of X L  and X L  . This completes the proof of assertion (ii), hence of Theorem 3.11. Remark 3.11.1. In passing, we observe that the condition concerning the kernel of the l-adic cyclotomic character on G k L  that appears in the final portion of Theorem 3.11, (i), is satisfied if k L  is either separably closed or algebraic over the finite field of cardinality p. We are now in a position to verify an absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields [cf. Theorem 3.12 below], which is one of the central open questions in anabelian geometry. Theorem 3.12 (Absolute version of the Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields). Let p , p be prime numbers; Σ Primes a subset of cardinality 2 that contains p and p ; K , K mixed characteristic local fields of residue characteristic p , p , respectively; X , X hyperbolic curves over K , K , respectively. Then the natural map Isom(X , X ) −→ OutIsom(Π X , Π X ) (Σ) (Σ) is bijective. Proof. First, we observe that any isomorphism of schemes between X and X necessarily lies over an isomorphism of fields between K and K . [Indeed, this × follows immediately by considering subgroups of the groups of units Γ(X , O X ), × Γ(X , O X ) whose unions with {0} are closed under addition.] Now Theorem 3.12 follows immediately by combining Theorems 2.17; 3.11, (i), (ii) [cf. also Remark 3.11.1], of the present paper with [AbsTopII], Corollary 2.9. Remark 3.12.1. Theorem 3.12 may be regarded as a complete affirmative resolu- tion of the absolute version of the Grothendieck Conjecture for hyperbolic curves over p-adic local fields in the geometrically pro-Σ case, where Σ Primes is a subset of cardinality 2 that contains the residue characteristic of the base field. On the other hand, the following questions remain open, to the authors’ knowledge, at the time of writing of the present paper: Question 1: Can one prove a geometrically pro-p version of the abso- lute Grothendieck Conjecture for hyperbolic curves over p-adic local fields? In this context, we observe that certain partial results in this direction are obtained in [Hgsh]. Question 2: Can one prove an absolute version of the Grothendieck Conjecture for hyperbolic curves over more general base fields? For instance, one may consider the case where the base fields are mixed 91 characteristic complete discrete valuation fields whose residue fields are algebraic over F p , i.e., a class of fields for which a relative ver- sion of the Grothendieck Conjecture for hyperbolic curves has been known for some time [cf. [AnabTop], Theorem 4.12]. Theorem 3.13 (Absolute version of the Grothendieck Conjecture for configuration spaces associated to arbitrary hyperbolic curves over p-adic local fields). Let p , p be prime numbers; K , K mixed characteristic local fields of residue characteristic p , p , respectively; X , X hyperbolic curves over K , K , respectively; n , n positive integers. Write X n (respectively, X n ) for the n -th (respectively, n -th) configuration space associated to X (respectively, X ). Then the natural map Isom(X n , X n ) −→ OutIsom(Π X , Π X ) n n is bijective. Proof. First, we observe that any isomorphism of schemes between X n and X n necessarily lies over an isomorphism of fields between K and K . [Indeed, this follows immediately by a similar argument to the argument applied in the proof of Theorem 3.12.] Now Theorem 3.13 follows immediately from a def routine argument via induction on n = n = n [cf. [AbsTopI], Theorem 2.6, (v); [HMM], Theorem A, (i), (ii)], by combining Theorem 3.12 of the present paper with the relative version of the Grothendieck Conjecture given in [LocAn], Theorem A. Next, we discuss the functorial behavior of the lengths of nodes of special fibers of compactified semistable models [cf. Definition 3.14 below] with respect to finite morphisms between compactified semistable models that extend finite étale Galois coverings of hyperbolic curves over mixed characteristic complete discrete valuation fields. Definition 3.14. Let K be a mixed characteristic complete discrete valuation field of residue characteristic p; X a hyperbolic curve over K; X a compactified semistable model with split reduction of X over O K ; e a node of X s . Recall that the completion of the local ring O X ,e at e is isomorphic to O K [[x, y]]/(xy a), where x, y denote indeterminates; a m K \ {0}. Then we shall refer to v p (a) as the length of e. [Note that the length of e is independent of the choice of a, as well as of the isomorphism O X ,e O K [[x, y]]/(xy a) over O K [cf. [Hur], §3.7].] 92 Proposition 3.15 (Functorial behavior of the lengths of nodes). Let K be a mixed characteristic complete discrete valuation field of residue character- istic p; X a hyperbolic curve over K; Y X a [connected] finite étale Galois covering of hyperbolic curves over K; Y a compactified semistable model of Y def over O K that is stabilized by G = Gal(Y /X). Write X for the compactified semistable model of X over O K obtained by forming the quotient of Y by the action of G on Y [cf. Proposition 2.3, (iv)]; f : Y X for the natural quotient morphism. Suppose that Y has split reduction, and that the natural action of G on Y s does not permute the branches of some node e Y of Y s . def Write e X = f (e Y ) for the node of X s determined by e Y [cf. Proposition 2.3, (iv)]; l X , l Y for the lengths of the nodes e X , e Y [relative to the compactified semistable models X , Y, respectively]. Then there exists a positive integer m such that l X = m · l Y . Moreover, the positive integer m may be computed as the cardinality of the decomposition subgroup [i.e., the stabilizer subgroup] of e Y in G. def Proof. Write S = Spec O K ; S log for the log scheme obtained by equipping S with the log structure determined by the closed point of S; X log , Y log for the log schemes over S log determined by the compactified semistable models X , Y, respectively [cf. the discussion of the subsection in Notations and Conventions entitled “Log schemes”]. Observe that f naturally determines a finite morphism f log : Y log X log of log schemes, hence a finite morphism  X ,e ) log ,  Y,e ) log −→ (Spec O (Spec O Y X  Y,e denote the completions of the respective normal local rings  X ,e , O where O X Y O X ,e X , O Y,e Y , and the superscripts “log” denote the log structures induced by the respective log structures of X log , Y log . Since Y is assumed to have split reduction, it follows [cf. the discussion of Definition 3.14] that  X ,e O K [[u 1 , u 2 ]]/(u 1 u 2 a), O X =  Y,e O K [[v 1 , v 2 ]]/(v 1 v 2 b), O Y = where u 1 , u 2 , v 1 , v 2 denote indeterminates; a, b m K \{0} are elements such that l X = v p (a), l Y = v p (b). Moreover, it follows immediately from the definitions of the log structures involved, together with the geometry of the irreducible  X ,e and Spec O  Y,e , that, after components of the special fibers of Spec O X Y possibly switching the indices {1, 2} of [either or both of] the pairs (u 1 , u 2 )  × and (v 1 , v 2 ), there exist positive integers m 1 , m 2 and units c 1 , c 2 O Y,e Y such that m 1 m 2 , and u 1 = c 1 · v 1 m 1 , u 2 = c 2 · v 2 m 2 , 93  Y,e via the natural injection O  X ,e → where we regard u 1 , u 2 as elements in O Y X  O Y,e Y . In particular, it holds that a = u 1 u 2 = c 1 c 2 v 1 m 1 v 2 m 2 = c 1 c 2 v 1 m 1 −m 2 b m 2 O K [[v 1 , v 2 ]]/(v 1 v 2 b). On the other hand, such a relation implies, in light of the well-known structure of the log structures involved, i.e., in effect, the geometry of the irreducible  X ,e and Spec O  Y,e , that m def components of the special fibers of Spec O = X Y m m 1 = m 2 , hence that a = c 1 c 2 b . In particular, we conclude that l X = m · l Y , as desired. Finally, the fact that m may be computed as the cardinality of the decomposition subgroup [i.e., the stabilizer subgroup] of e Y in G follows immediately from the fact that the generic degree of the [finite, generically  Y,e Spec O  X ,e is [easily computed, via the explicit étale] morphism Spec O Y X presentations of O Y,e Y , O X ,e X given above, to be] m. This completes the proof of Proposition 3.15. Next, we apply Proposition 3.15, together with the theory of p-adic arith- metic cuspidalizations developed in [Tsjm], §2, to prove that the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature [cf. [Tsjm], Remark 2.1.2] in fact coincide. Theorem 3.16 (Equality of various p-adic versions of the Grothendieck- def -Teichmüller group). Write X = P 1 Q \ {0, 1, ∞}; p GT Out(Π X ) for the Grothendieck-Teichmüller group [cf. [CmbCsp], Remark 1.11.1]; GT M GT (⊆ Out(Π X )) for the metrized Grothendieck-Teichmüller group [cf. [CbTpIII], Remark 3.19.2]; def GT tp = GT Out(Π tp p X ) Out(Π X ) [cf. the subsection in Notations and Conventions entitled “Fundamental groups”; [Tsjm], Definition 2.1]. Then the natural inclusion GT M GT tp p of subgroups of GT is an equality. In particular, it holds that GT M = GT p = GT G = GT tp p [cf. [Tsjm], Remark 2.1.2]. 94 Proof. First, we recall that there exists a natural surjection φ : GT tp p  G Q p whose restriction to G Q p is the identity automorphism [cf. [Tsjm], Corollary B, as well as Remark 3.16.1 below]. Thus, since G Q p GT M GT tp p , it suffices to prove that Ker(φ) GT M . Let σ Ker(φ). Fix a lifting σ̃ Aut(Π tp X ) of will always denote the Primes- σ. [Here and in the following discussion, Π tp (−) tempered fundamental group of (−).] Then it follows immediately from the construction of φ [cf. the discussion, in the proof of [Tsjm], Corollary 2.4, of the two paragraphs following the proof of Claim 2.4.B; the discussion, in the proof of [HMT], Theorem 4.4, of the observa- tion immediately following the statement of Claim 4.4.A; [NCBel], Corollary 1.2] that, for any finite subset S Q \ {0, 1} X(Q p ), σ̃ lifts to an automorphism def of Π tp X S [where we write X S = X \ S] with respect to the natural surjection tp Π tp X S  Π X determined [up to composition with an inner automorphism] by the natural open immersion X S → X. Next, let ψ Y : Y X be a connected finite étale covering over Q p . Write ψ Z : Z X for the Galois closure of ψ Y ; Z for the compactified stable model of σ Z over O Q p ; ψ Y σ : Y σ X, ψ Z : Z σ X for the connected finite étale coverings tp tp tp over Q p that correspond to the open subgroups σ̃(Π tp Y ) Π X , σ̃(Π Z ) Π X , respectively. For each finite subset S Q \ {0, 1} X(Q p ), write def Z S (respectively, Z S σ ) for the compactified stable model of Z S = Z \ −1 σ −1 (S) (respectively, Z S σ = Z σ \ Z ) (S)) over O Q p ; ψ Z def X S (respectively, Y S , Y S σ ) for the compactified semistable model of X S (respectively, Y S = Y \ ψ Y −1 (S), Y S σ = Y σ \ Y σ ) −1 (S)) obtained by forming the quotient of Z S (respectively, Z S , Z S σ ) via the natural action of Gal(Z/X) (respectively, Gal(Z/Y ), Gal(Z σ /Y σ )) [cf. Proposition 2.3, (iv)]. def def Next, observe that there exists a finite subset T Q \ {0, 1} X(Q p ) such that the natural action of Gal(Z/X) on (Z T ) s does not permute any branches of nodes, and X T , Y T , Y T σ are the respective compactified stable models of X T , Y T , Y T σ over O Q p . Then since σ̃ lifts to an automorphism of Π tp X T [cf. the above discussion], hence to an isomorphism tp tp tp tp tp Π tp Z × Π tp Π X T = Π Z T Π Z σ = Π Z σ × Π tp Π X T , T X 95 X it follows immediately from Proposition 2.3, (iv), together with [SemiAn], Corol- lary 3.11, that σ̃ induces a commutative diagram of semi-graphs Γ Z T −−−−→ Γ Z T σ   Γ X T Γ X T , where Γ (−) denotes the dual semi-graph associated to (−) s , compatible with the respective natural actions of Gal(Z/X), Gal(Z σ /X). Thus, we conclude from Proposition 3.15 that the isomorphism Γ Z T Γ Z T σ of dual semi-graphs is compatible with the respective metric structures [cf. [CbTpIII], Definition 3.5, (iii)]. On the other hand, σ̃ also induces a commutative diagram of semi-graphs Γ Z T −−−−→ Γ Z T σ   Γ Y T −−−−→ Γ Y T σ compatible with the respective natural actions of Gal(Z/Y ), Gal(Z σ /Y σ ) [cf. Proposition 2.3, (iv); [SemiAn], Corollary 3.11]. Thus, since the isomorphism Γ Z T Γ Z T σ of dual semi-graphs is compatible with the respective metric struc- tures, we conclude from Proposition 3.15 again that the isomorphism Γ Y T Γ Y T σ of dual semi-graphs is also compatible with the respective metric struc- tures. Finally, it follows immediately from the well-known theory of pointed stable curves and contraction morphisms that arise from eliminating cusps, as exposed in [Knud] [cf. also Remark 2.1.4], that this implies that, if we write Y, Y σ for the respective compactified stable models of Y , Y σ over O Q p , then the isomorphism Γ Y Γ Y σ of dual semi-graphs induced by σ̃ [cf. [SemiAn], Corollary 3.11] is compatible with the respective metric structures. Thus, we conclude from [CbTpIII], Definition 3.7, (ii); [CbTpIII], Remark 3.19.2, that GT M = GT tp p . This completes the proof of Theorem 3.16. Remark 3.16.1. Here, we recall that one of the key ingredients in the proof of [Tsjm], Corollary B, is the theory of resolution of nonsingularities developed in [Lpg1]. As a corollary, we obtain the following affirmative answer to the question posed in the discussion immediately preceding Theorem E in [CbTpIII], Intro- duction: Corollary 3.17 (Commensurable terminality of various p-adic versions of the Grothendieck-Teichmüller group). We maintain the notation of Theorem 3.16. Then GT M = GT p = GT G = GT tp p is commensurably terminal M in GT, i.e., the commensurator C GT (GT ) of GT M in GT is equal to GT M . 96 Proof. It follows immediately from Theorem 3.16, together with [CbTpIII], The- orem E, that GT M C GT (GT M ) GT G = GT M . Thus, we conclude that C GT (GT M ) = GT M , as desired. Proposition 3.18 (Reconstruction of the subset of Q p -rational points def from the p-adic Grothendieck-Teichmüller group). Write X = P 1 Q \ p {0, 1, ∞}; Π tp X for the Primes-tempered fundamental group of X. Then the subset X(Q p ) X(C p ), where we think of “X(C p )” as the set reconstructed from Π tp X in Corollary 3.10, may be reconstructed, in a purely combinatorial/group- theoretic way, from the data tp tp tp X , GT p Out(Π X )) tp consisting of the underlying topological group of Π tp X and the subgroup GT p tp Out(Π X ) as the subset of elements fixed by some open subgroup of GT tp p . Moreover, this reconstruction procedure is functorial with respect to isomor- phisms of topological groups for which the induced isomorphism on “Out(−)” preserves the given subgroup of “Out(−)”. Proof. First, we observe that it follows immediately from the existence of the natural homeomorphism “θ X  of Proposition 2.3, (viii), together with the defi-  tor [cf. Definition 2.2, (vi)], that the subset X(Q p ) X(C p ) nition of “VE( X) is dense in X(C p ), and that the natural action of GT tp p on X(C p ) is via self- homeomorphisms of X(C p ) [cf. Corollary 3.10 and its proof; Corollary 3.16]. Thus, since the natural action of GT tp p on X(Q p ) factors through the surjection tp GT p  G Q p [cf. [Tsjm], Corollary B, and its proof], we conclude that the tp natural action GT tp p on X(C p ) factors through this surjection GT p  G Q p , and hence [cf. [Tate], §3.3, Theorem 1] that the subset X(Q p ) X(C p ) may be characterized as the subset of elements fixed by some open subgroup of GT tp p . This completes the proof of Proposition 3.18. Finally, we apply the theory of resolution of nonsingularities and point- theoreticity [cf., especially, Corollary 2.5, (i); Corollary 3.10], together with the theory of metric-admissibility developed in [CbTpIII], §3, to construct certain arithmetic cuspidalizations of the [Primes-]tempered fundamental groups of hy- perbolic curves over Q p equipped with “proj-metric structures” [cf. Definition 3.19 below]. Definition 3.19. Let Σ Primes be a subset of cardinality 2 that con- tains p; K a mixed characteristic complete discrete valuation field of residue characteristic p; X a hyperbolic curve over K. Write Π tp X for the Σ-tempered fundamental group of X. For each open subgroup Π Π tp X of finite index, write 97 X Π for the compactified stable model over O K of the hyperbolic curve over K corresponding to the open subgroup Π Π tp X ; Γ Π for the dual graph associated to (X Π ) s ; μ Π for the metric structure on Γ Π associated to X Π , considered up to multiplication by a constant Q × [cf. [CbTpIII], Definition 3.5, (iii)]. Then we shall refer to μ Π as the proj-metric structure on Γ Π . We shall refer to the collection of data of proj-metric structures Π } associated to the charac- teristic open subgroups Π tp X } of finite index as the proj-metric structure tp on Π X . Theorem 3.20 (Construction of certain arithmetic cuspidalizations of geometric tempered fundamental groups). Let K be a mixed characteris- tic complete discrete valuation field of residue characteristic p; X a hyperbolic  a universal pro-Primes covering of X. Suppose that X sat- curve over K; X isfies Primes-RNS. Write Ω for the p-adic completion of K. For n 2 an def integer, write X n for the n-th configuration space associated to X; Π 1 = Π X ; def Π n = Π X n ; Π 2/1 for the kernel of the natural surjection Π 2  Π 1 induced by the first projection X 2 X, where we regard Π 2 as a quotient of Π n via the projection X n X 2 to the first two factors; Π tp 1 for the Primes-tempered fundamental group of X; def (Out(Π n ) ⊇) Out(Π n ) tp = Out gF n ) × Out(Π 1 ) Out(Π tp 1 ), def (Out(Π n ) ⊇) Out gFC n ) = Out gF n ) Out FC n ) (⊆ Out(Π 1 )), def (Out(Π n ) ⊇) Out gFC n ) M = Out gF n ) Out FC n ) M (⊆ Out(Π tp 1 ) Out(Π 1 )) [cf. [HMM], Definition 2.1, (iv); [CbTpI], Theorem A, (i); [CbTpIII], Propo- sition 3.3, (iv); [CbTpIII], Definition 3.7, (i), (ii), (iii); [NodNon], Theorem B]; VE(Π tp Π tp 1 ), 1 (Ω)  and X(Ω)”  for the respective sets “VE( X)” equipped with their natural actions tp by Aut(Π 1 ) and Π 1 constructed in [the proof of ] Corollary 2.5, (i), and Corol- tp lary 3.10 from the underlying topological group of Π tp 1 . Let VE(Π 1 ). Then the following hold: (i) One may construct an “arithmetic cuspidalization” of Π tp 1 associated to from the data consisting of the topological group Π n equipped with the quotients Π n  Π 2  Π 1 and a topology [i.e., the tempered topology] on the subquotient Π n  Π 2  Π 1 Π tp 1 98 in a fashion that is functorial with respect to isomorphisms of this data [in the evident sense] as follows: Observe that the subgroup Out(Π n ) tp Out(Π n ) may be constructed from the given data [cf. [HMM], Theorem A, (ii)]. Write tp tp out tp n D Π 1  Out(Π n ) tp = Aut(Π tp 1 ) × Out(Π tp ) Out(Π n ) 1 [cf. [CbTpIII], Proposition 3.3, (i), (ii); [MT], Proposition 2.2, (ii)] for the stabilizer subgroup of x̃. Note that there exists a natural exact sequence [that may be constructed from the given data] out out tp tp 1 −→ Π 2/1 −→ 2 × Π 1 Π tp −→ Π tp −→ 1. 1 )  Out(Π n ) 1  Out(Π n ) Thus, by pulling-back the above exact sequence via the inclusion n D tp out tp Π tp 1  Out(Π n ) , we obtain an exact sequence out 1 −→ Π 2/1 −→ Π 2/1  n D tp −→ n D tp −→ 1. out We shall refer to Π 2/1  n D tp as the [n-th] arithmetic cuspidalization of Π tp 1 associated to x. (ii) Write n-alg Π tp Π tp 1 (Ω) 1 (Ω) for the subset of elements ξ Π tp 1 (Ω) whose Π 1 -orbit Π 1 · ξ is stabilized by some open subgroup of Out gFC n ) M (⊆ Out(Π tp 1 )) [cf. Remark 3.20.1 n-alg Π tp below]. Suppose that arises from an element Π tp 1 (Ω) 1 (Ω), which, by a slight abuse of notation, we shall also denote by x̃. Write def X x = X Ω \ {x}, where x X(Ω) denotes the element determined by x̃. Then the [Primes-]tempered fundamental group Π tp X x (⊆ Π 2/1 ) of X x [where we identify Π 2/1 with Π X x ], together with the proj-metric structure on Π tp X x , may be reconstructed, in a purely combinatorial/group- theoretic way, from the following data the topological group Π n equipped with the quotients Π n  Π 2  Π 1 and a topology [i.e., the tempered topology] and proj-metric structure on the subquotient Π n  Π 2  Π 1 Π tp 1 ; the subgroup Out gFC n ) Out(Π n ) [cf. Remark 3.20.2 below] in a fashion that is functorial with respect to isomorphisms of this data [in the evident sense]. 99 Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii). Write M def = n D out gFC Π tp n ) M 1  Out n D tp out tp (⊆ Π tp 1  Out(Π n ) ). In particular, it follows immediately from assertion (i), together with the various definitions involved, that one may construct, from the given data, n D M , together with the natural outer action of n D M on Π 2/1 . Let Π 2/1 Π 2/1 be an open subgroup that is normal in Π 2 ; l a prime number = p. Write K K tm (⊆ K) def tm /K ur ); Π 2/1  Π 2/1 for for the maximal tame extension of K; G tm K = Gal(K the maximal almost pro-l quotient associated to [i.e., “with respect to”] Π 2/1 Π 2/1 [cf. [CbTpIII], Definition 1.1]. Let us assume further that the quotient Π 2  Π 2 /Ker(Π 2/1  Π 2/1 ) is F-characteristic [cf. [CbTpIII], Definition 2.1, (iii)]. Note that this implies that, relative to the identification of Π 2/1 with Π X x [cf. the statement of assertion (ii)], the natural Π 2/1 -outer action of G K on Π 2 [which is well-defined after possibly replacing K by a suitable finite extension field of K] induces a natural outer action of G K on Π 2/1 . Thus, in order to complete the proof of assertion (ii), it suffices, in light of the argument given in the proof of [CbTpIII], Theorem 3.9 [cf., especially, the equivalence stated in the final display of the proof of [CbTpIII], Theorem 3.9], to reconstruct, after possibly replacing K by a suitable finite extension field of K, the image of G tm K in Out(Π 2/1 ) via the natural outer representation. Next, we verify the following assertion:  →) Y X be a [connected] finite étale Galois Claim 3.20.A: Let ( X covering over K. Then there exist a compactified semistable model  Y of Y over O K and a [connected] finite étale Galois covering ( X ) Z X over K that dominates Y X and satisfies the following conditions: The compactified stable model Z of Z over O K dominates Y. The component of the VE-chain corresponding to Z is an irreducible component of Z s that maps to a smooth closed point Y s that is not a cusp. Indeed, since X satisfies Primes-RNS, Claim 3.20.A follows immediately from the fact that the VE-chain arises from an element Π tp 1 (Ω) that is of type 1, hence weakly verticial [cf. Proposition 2.4, (v); Remark 3.1.1; Proposition 3.3, (ii); Proposition 3.4, (iii); the theory of pointed stable curves, as exposed in [Knud]]. Next, we observe that it follows from [CbTpIII], Proposition 2.3, (ii) [cf. conditions (a), (b), (c) below]; [CbTpIII], Corollary 2.10 [cf. condition (c) be- low], together with Claim 3.20.A [cf. condition (b) below], that, after possibly replacing Π 2/1 Π 2/1 by a smaller open subgroup that satisfies the same con- ditions as Π 2/1 , there exist F-characteristic SA-maximal almost pro-l quotients Π 2  Π 2 , Π 2  Π ∗∗ 2 satisfying the following conditions: 100 (a) The F-characteristic SA-maximal almost pro-l quotient Π 2  Π ∗∗ 2 dom- inates the F-characteristic SA-maximal almost pro-l quotient Π 2  Π 2 . In particular, we obtain a commutative diagram of profinite groups ∗∗ ∗∗ 1 −−−−→ Π ∗∗ 2/1 −−−−→ Π 2 −−−−→ Π 1 −−−−→ 1    1 −−−−→ Π 2/1 −−−−→ Π 2 −−−−→ Π 1 −−−−→ 1, ∗∗ where the quotients Π 1 , Π ∗∗ 1 of Π 1 induced by Π 2 , Π 2 are the center-free [cf. [CbTpIII], Proposition 1.7, (i)] maximal almost pro-l quotients of Π 1 associated to normal open subgroups of Π 1 ; the quotients Π 2/1 , Π ∗∗ 2/1 of Π 2/1 induced by Π 2 and Π ∗∗ are the center-free [cf. [CbTpIII], Proposition 2 1.7, (i)] maximal almost pro-l quotients of Π 2/1 associated to normal open subgroups of Π 2/1 ; the vertical arrows denote surjective homomorphisms. (b) Fix a normal open subgroup Π Y Π 1 whose associated maximal almost pro-l quotient coincides with Π 1  Π 1 . Then there exists a normal open subgroup Π Z Π 1 such that: It holds that Π Z Π Y . [In particular, the maximal almost pro-l quotient associated to the normal open subgroup Π Z Π 1 dominates the maximal almost pro-l quotient Π 1  Π 1 .] The maximal almost pro-l quotient Π 1  Π ∗∗ 1 dominates the maximal almost pro-l quotient associated to the normal open subgroup Π Z Π 1 .  →) Y X, ( X  →) Z X for the respective [connected] Write ( X finite étale Galois coverings over K associated to the normal open subgroups Π Y Π 1 , Π Z Π 1 . Then there exists a compactified semistable model Y of Y over O K such that the compactified stable model Z of Z over O K dominates Y, and, moreover, the component of the VE-chain corresponding to Z is an irreducible component of Z s that maps to a smooth closed point Y s that is not a cusp. FC ∗∗ ∗∗ (c) Every element Out FC ∗∗ 2  Π 2 ) Ker(Out 2 ) Out(Π 1 )) [cf. [CbTpIII], Definition 2.1, (viii)] induces the trivial outer automorphism of Π 2 . Write D ∗∗ out out for the image of n D M (⊆ Π 1  Out gFC n ) Π 1  Out gFC 2 )) [where the second inclusion follows from [NodNon], Theorem B] via the natural homomor- out out FC phism Π 1  Out gFC 2 ) Π ∗∗ ∗∗ 1  Out 2 ); out FC ∗∗ ρ ∗∗ : D ∗∗ Π ∗∗ ∗∗ 1  Out 2  Π 2 ) −→ Out(Π 1 ) 101 for the natural composite homomorphism. Next, we verify the following assertion: Claim 3.20.B: There exists an open subgroup D ∗∗ D ∗∗ such that every element D ∗∗ Ker(ρ ∗∗ ) induces the trivial outer action on Π 2/1 . Indeed, observe that since Π ∗∗ 1 is center-free [cf. condition (a)], the natural homomorphism out ∗∗ ∗∗ D ∗∗ −→ Π ∗∗ 1  Out(Π 1 ) = Aut(Π 1 ) induces a natural homomorphism φ : Ker(ρ ∗∗ ) −→ Π ∗∗ 1 . Thus, it follows immediately from the final portion of condition (b) [cf. also the proof of Claim 3.20.A; our assumption that X satisfies Primes-RNS; the portion of Proposition 3.5, (i), concerning the weakly verticial case], together with the various definitions involved [cf., especially, the definition of D ∗∗ ], that the image of the natural composite homomorphism φ Ker(ρ ∗∗ ) −→ Π ∗∗ 1  Π 1 is finite. In particular, there exists an open subgroup D ∗∗ D ∗∗ such that every element D ∗∗ Ker(ρ ∗∗ ) induces the trivial automorphism of Π 1 . On the other hand, this implies, in light of condition (c), that every element D ∗∗ Ker(ρ ∗∗ ) induces the trivial outer automorphism of Π 2 . Thus, since Π 1 is center-free [cf. condition (a)], we conclude that every element D ∗∗ Ker(ρ ∗∗ ) induces the trivial outer automorphism of Π 2/1 . This completes the proof of Claim 3.20.B. Next, let us observe that, by applying the argument given in the proof of Theorem 3.9 [cf., especially, the equivalence stated in the final display of the proof of [CbTpIII], Theorem 3.9], together with condition (a), we conclude that, after possibly replacing K by a suitable finite extension field of K, one tm may reconstruct, from the proj-metric structure on Π tp 1 , the image I of G K in Out(Π ∗∗ ) via the natural outer representation. On the other hand, observe 1 that ρ ∗∗ (D ∗∗ ) contains an open subgroup of I [cf. Remark 3.20.1 below]. Thus, since D ∗∗ Ker(ρ ∗∗ ) induces the trivial outer action on Π 2/1 [cf. Claim 3.20.B], by considering the outer action of D ∗∗ ∗∗ ) −1 (I) on Π 2/1 , we conclude that, after possibly replacing K by a suitable finite extension field of K, one may reconstruct the image of G tm K in Out(Π 2/1 ), as desired. This completes the proof of assertion (ii), hence of Theorem 3.20. 102 Remark 3.20.1. Suppose that we are in the situation of Theorem 3.20, (ii). Then it follows immediately from the definitions [cf. also the natural homomorphism G K Out gFC n ) M [which is well-defined after possibly replacing K by a suit- able finite extension field of K]; the proof of Theorem 3.11, (ii)] that the subset tp tp n-alg Π tp Π tp 1 (Ω) 1 (Ω) is contained in the subset Π 1 (K) Π 1 (Ω) correspond-  constructed in Corollary 3.10. ing to the K-rational points of the set X(Ω)” Moreover, it follows immediately from Proposition 3.18 [cf. also Theorem 3.16; [HMM], Corollaries B, C] that the inclusion n-alg Π tp Π tp 1 (Ω) 1 (K) is in fact an equality in the case where X = P 1 Q \ {0, 1, ∞}. It is not clear to p the authors, however, at the time of writing of the present paper whether or not the inclusion of the above display is an equality in general. Remark 3.20.2. Suppose that we are in the situation of Theorem 3.20, (ii). 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